graph transformations calculator

This Laplace function can be solved using an algebraic equation. Share your findings on the discussion board. In general, this describes the horizontal translations; if \(h\) is any positive real number: Begin with a basic cubing function defined by \(f(x)=x^{3}\) and shift the graph \(4\) units to the right. Summary of Transformations To graph Draw the graph of f and: Changes in the equation of y = f(x) Vertical Shifts y = f (x) + c Begin with the basic function defined by f(x) = x and shift the graph up 4 units. Compare y = x 2 and y = x 2 + k where k is any integer, positive or negative. How Many Fluid Ounces in a Gallon of Water? WebTransformations of Functions DESMOS Online Calculator Exploration Activity. \(y = x^{3}\) ; Shift right \(2\) units; domain: \(\); range: \(\), 19. Algebra Examples Free graphing calculator instantly graphs your math problems. Algebra. WebGet the free "Reflection Calculator MyALevelMathsTutor" widget for your website, blog, Wordpress, Blogger, or iGoogle. Both electronic and mechanical engineering benefit from them. \(y = \frac{1}{x}\); Shift left \(1\) unit and down \(2\) units; domain: \((, 1) (1, )\); range: \((, 2) (2, )\), 27. They aid in the analysis of variables that, when altered, create the intended results. To accomplish this task, simply press [CTRL] [CLICK to grab the graph and then use the Touchpad keys to perform a transformation. Step 2: Click the blue arrow to submit and see your result! It is often the case that combinations of translations occur. example. It is obtained from the graph of f(x) = 0.5x3+1 by reflecting it in the y-axis. If 0 < c < 1, (a proper fraction) then the graph is stretched horizontally. Plot y = x 2 4, y = x 2 1. As an example, create the graph of y = x2. When you let go of the slider it goes back to the middle so you can zoom more. Note that this is like erasing the part of the graph to the left of the -axis and reflecting the points from the right of the -axis over to the left. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Step 2: Now click the button Plot Graph to get the graph. WebExample 4: applying a reflection in the y- axis. In most cases, we solve differential equations using Laplace transformations. Find more Education widgets in Wolfram|Alpha. However, the bilateral Laplace transform is defined as: However, the other way to represent the bilateral Laplace transform is B{F}, instead of F. Everything youre supposed to graph independently is predicated on a more fundamental graph (parent function) that you MUST know. However, the integral transform of a given derivative function is the laplace transform. inverse sine (arcsine) of a value or expression, inverse cosine (arccos) of a value or expression, inverse tangent (arctangent) of a value or expression, Hyperbolic cosine (cosh) of a value or expression, Hyperbolic tangent (tanh) of a value or expression, e (the Euler Constant) raised to the power of a value or expression, The natural logarithm of a value or expression, The base10 logarithm of a value or expression. Usage To plot a function just type it into the function box. WebThe procedure to use the transformations calculator is as follows: Step 1: Enter any function in the input field. Exercise 4 Finding the Equation of a Given Graph. (If you have a second equation use a semicolon like y=2x+1 ; y=x+3) Press WebInteractive, free online graphing calculator from geogebra: Transformations with the desmos graphing calculator. The function \(h\) is not as steep as the basic squaring function and appears to have been stretched horizontally. Get detailed step-by-step solutions. powered by "x" x "y" y "a" squared a 2 "a" Superscript Transformations: Inverse of a Function. Match the graph to the given function defintion. Download full solution. Calculate it! Sketch the graph of \(g ( x ) = - | x | + 3\). Our online transformation calculator is simple and straightforward to use, with the result appearing in a matter of seconds. However, the Laplace transform of f(t), F(s), can then be written as, Provided that the integral exists. Matrixes can be used to express the above transformations (rotation, reflection, scaling, and shearing). WebThe graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis. Determine whether the transformation is a translation or reflection. 5. Use the Function Graphing Rules to find the equation of the graph in green and list the rules you used. Pre-Algebra. When signals are conveyed over the phone, for example, first we transform them into a time-varying wave before we superimpose it on the medium. \\ { y = - ( x + 5 ) ^ { 2 } } \quad\quad\:\:\:\color{Cerulean}{Horizontal\: shift\: left\: 5\: units.} (19) $2.00. Multiplying the values in the domain by \(1\) before applying the function, \(f(x)\), reflects the graph about the \(y\)-axis. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. It is obtained from the graph of f(x) = 0.5x3+1 by reflecting it in the y-axis. Calculate it! Visit Mathway on the web. Download free on. 5.0. As a result, the typical unilateral Laplace transform is transformed into a specific case of the Bilateral Laplace transform, with the function definition increased by the Heaviside step function. Solve Now. Its similar to an on/off switch. This fascinating concept allows us to graph many other types of functions, like square/cube root, exponential and logarithmic functions. All graphs/images were created with GeoGebra. We can also move them along the xy plane. Statistics: Linear Regression. Christle Clear Math Resources. Calculator for Transformations: Definition of Laplace Transformation. Step 1: Enter the expression you want to evaluate. 560+ Math Tutors. Graphing Calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Loading Graph Transformations. (If you have a second equation use a semicolon like y=2x+1 ; y=x+3) Press For example, we can multiply the squaring function \(f(x) = x^{2}\) by \(4\) and \(\frac{1}{4}\) to see what happens to the graph. WebExplore math with our beautiful, free online graphing calculator. Note that this is the opposite of what you might expect. y = f (x). Loading Graph Transformations. Vector transformation works the same way. \(y = \frac{1}{x}\); Shift up \(5\) units; domain: \((, 0) (0, )\); range: \((, 1) (1, )\), 25. \(y = |x|\); Shift right \(1\) unit and down \(3\) units; domain: \(\); range: \([3, )\), 13. WebCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. WebWe can think graphs of absolute value and quadratic functions as transformations of the parent functions |x| and x. Sketch the graph and state the coordinate of the image of point P P on the graph y=f (-x). WebJust like Transformations in Geometry, we can move and resize the graphs of functions Let us start with a function, in this case it is f (x) = x2, but it could be anything: f (x) = x2 Here are some simple things we can do to move or scale it on the graph: We can move it up or down by adding a constant to the y-value: g (x) = x2 + C In general, transformations in y-direction are easier than transformations in x-direction, see below. Calculate it! Example 2.5.1: Sketch the graph of g(x) = x + 4. If a number is being added outside of parentheses, shift it up by that amount, or if a number is being removed, shift it down by that amount (ex: f(x) = x22 moves down 2). 582+ Math Consultants 13 Years in business powered by "x" x "y" y "a" squared a 2 "a" Superscript Transformations: Inverse of a Function. Vector transformation works the same way. example. Mathway. Use rectangular, polar, cylindrical, or spherical coordinates. Multiplying a function by a negative constant, \(f(x)\), reflects its graph in the \(x\)-axis. Find more Education widgets in Wolfram|Alpha. We may perform the laplace transforms of various functions in order to analyse the control system. However, the constants a and b are used in this equation. example. If a negative exists inside parenthesis, horizontally reflect over the y-axis (all x-values become negative) and factor out the negative. However, it worked on it until 1809, when he began utilising infinity as an integral condition, revealing the true power of the Laplace transform. If a positive constant is subtracted from a function, \(f(x) k\), the graph will shift down. Download free in Windows Store. Use "x" as the variable like this: Examples: sin (x) 2x3 cos (x^2) (x3) (x+3) Zooming and Re-centering WebBasic Math. WebThe following steps make graphing transformations so easier. 5/5 - (5 votes) Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. For example, lets say you had the function f: x x 2 and you wanted to transform (map) the number 2. Transformation Calculator Differential Equation, Transformation Calculator Bilateral Laplace, 3. By a ratio of two, 2 is expanded horizontally. WebGraph Transformations 1 ( AGG) Investigate the transformations of the graph y = f (x) + b, and how this affects the graph of y = f (x). Reflect Over X-Axis or Y-Axis 3. Download free on Amazon. If 0 < c < 1, (a proper fraction) then the graph is stretched horizontally. Just add the transformation you want to to. Reflect over the x-axis, or vertically, if there is a negative outside parentheses (all the y-values become negative). \(y = x^{2}\); Shift right \(5\) units and up \(2\) units; domain: \(\); range: \([2, )\), 9. PDF. You can also save your work as a URL (website link). Answer: Figure 2.5.3. Calculus. Download free on iTunes. Here, we are transforming the function y = f (x) to y = a f (b (x + c)) + d. Step 1: Note down some coordinates on the original curve that define its shape. Statistics: Anscombe's Quartet. How do you find the transformation of a graph? Graphing. WebThere are 4 main types of graph transformation that we will cover. WebTransforming Without Using t-charts (steps for all trig functions are here). powered by "x" x "y" y "a" squared a 2 "a" Superscript Transformations: Inverse of a Function. Example 2.5.1: Sketch the graph of g(x) = x + 4. Plug in a couple of your coordinates into the parent function to double check your work, Transformation Calculator Inverse Laplace, Transformation Calculator Convolution Integrals, Applications of Transformation Calculator, Disadvantages of the Transformation Calculator, Frequently Asked Questions on Transformation Calculator. Download free on iTunes. As an example, create the graph of y = x2. Primary full-load current, I 1 = (50 1000 / 4000) = 12.5 A. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Step 2: Multiply each x -value by 1 2. example. If the factor \(a\) is a nonzero fraction between \(1\) and \(1\), it will stretch the graph horizontally. The following Laplace transform table helps to solve the differential equations for different functions: A well-known mathematical technique for solving a differential equation is the Laplace transform. Transformation Calculator Graph 1. Shift (Translate) Vertically or Horizontally 4. However, the function in the time domain is changed into a Laplace function in the frequency domain using the Laplace Transform method. \(f ( x ) = \frac { 1 } { x + 8 } + 4\). WebInteractive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! WebThe following steps make graphing transformations so easier. WebThe Transformations Calculator is an online tool that finds the Laplace Transform of any given function in the time domain and presents the result in the frequency domain. Step 3: Finally, the graph of the quadratic function will be displayed in the new window. Step 1: Enter the expression you want to evaluate. What is Transformation in quadratic equations? Step 2: Now click the button Plot Graph to get the graph. What is a transformation in a math example? Trigonometry. WebThe procedure to use the transformations calculator is as follows: Step 1: Enter any function in the input field. Just add the transformation you want to to. In other words, if c > 1, then the graph is compressed. Secondary full-load current, I 2 = (50 1000 / 400) = 125 A. b) Turns Ratio = N 1 / N 2 = V 1 / V 2 = (4000 / 400) = 10. Take care to shift the vertical asymptote from the y-axis 5 units to the right and shift the horizontal asymptote from the x-axis up 3 units. The basic shape of the graph will remain the same. Use the transformations to graph the following functions. Step 2: Click the blue arrow to submit and see your result! Begin by evaluating for some values of the independent variable \(x\). We will also go over the definition of the Laplace transform, its formula, properties, the Laplace transform table, and its applications in depth in this post. Primary full-load current, I 1 = (50 1000 / 4000) = 12.5 A. Log InorSign Up. Graphing Calculator. Precalculus. Define the rigid transformations and use them to sketch graphs. example. We named the Laplace transform (or Laplace technique) after Pierre Simon De Laplace, a prominent French mathematician (1749-1827). Use the Function Graphing Rules to find the equation of the graph in green and list the rules you used. WebThe Transformations Calculator is an online tool that finds the Laplace Transform of any given function in the time domain and presents the result in the frequency domain. Summary of Transformations To graph Draw the graph of f and: Changes in the equation of y = f(x) Vertical Shifts y = f (x) + c Functions can get pretty complex and go through transformations, like reflections along the x- or y-axis, shifts, stretching and shrinking, making the usual graphing techniques difficult. You can also add, subtraction, multiply, and divide and complete any arithmetic you need. Step 3: Finally, the Laplace transform of the given function will be displayed in the new window. WebFunction Grapher is a full featured Graphing Utility that supports graphing up to 5 functions together. We move every point in the shape in the same direction and at the same distance. Find more Education widgets in Wolfram|Alpha. Algebra Examples Free graphing calculator instantly graphs your math problems. Use "x" as the variable like this: Examples: sin (x) 2x3 cos (x^2) (x3) (x+3) Zooming and Re-centering If you are one of We obtained the Laplace transformation of f(t) as a result of this procedure, that we denote with F. (s). The diagram shows the graph of y=f (x) y = f (x) and a point on the graph P (2,5). WebExplore equations of transformed curves. Surface Of Revolution Calculator < Math Calculators List > X And Y Intercepts Finder Calculator. This occurs when we add or subtract constants from the \(x\)-coordinate before the function is applied. Download full solution. A vertical translation59 is a rigid transformation that shifts a graph up or down relative to the original graph. Loading Untitled Graph. Download free on Google Play. Web8 12. WebTransforming Without Using t-charts (steps for all trig functions are here). Statistics: Linear Regression. A differential equation can be used to model an electrical, mechanical, thermal, hydraulic, or other dynamic control system. Returns the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer. However, the transforms are employed in the study and analysis of systems such as ventilation, heating, and air conditioning. This type of non-rigid transformation is called a dilation62. Exercise 4 Finding the Equation of a Given Graph. This fascinating concept allows us to graph many other types of functions, like square/cube root, exponential and logarithmic functions. Importantly, we can extend this idea to include transformations of any function whatsoever! Finite Math. WebIn this video, I show you how to transform graphs using the list feature on the TI-84 Plus CE graphing calculator. Math Calculator. WebTransforming Without Using t-charts (steps for all trig functions are here). that coefficient to see the graph compress horizontally (the opposite of what youd expect). WebGraph transformation calculator. \end{array}\). If you are one of WebHow to Use the Quadratic Function Calculator? Web8 12. Here, we are transforming the function y = f (x) to y = a f (b (x + c)) + d. Step 1: Note down some coordinates on the original curve that define its shape. Graph Transformations 3 ( AGG) It became interested in LaGranges work 38 years later, in 1782, when he started up where Euler left off. Working out the vector transformation is equivalent to working out a function and involves some basic math. 4.9. As a result, we can only solve differential equations with proper constants using this method. WebGraph transformations Given the graph of a common function, (such as a simple polynomial, quadratic or trig function) you should be able to draw the graph of its related function. Vertical and Horizontal Stretches/Compressions 5. Also, it had a stroke of genius in 1785 that forever transformed the way we solve differential equations. WebExplore math with our beautiful, free online graphing calculator. Log InorSign Up. WebExplore equations of transformed curves. Shift (Translate) Vertically or Horizontally, 4. P (2,5). Read Also: Slant Asymptote Calculator Online. We may trace the history of the Laplace Transform all the way back to 1744. Similarly, if a whole number coefficient is enclosed in parenthesis, multiply the x-values of all points by the inverse of (!) Aimed at C1 students. Compare the graph of \(g\) and \(h\) to the basic square root function defined by \(f(x)=\sqrt{x}\), shown dashed in grey below: The first function \(g\) has a negative factor that appears inside the function; this produces a reflection about the \(y\)-axis. Mathway. Step 1: Identify the transformation on the parent graph, f. y = f ( x) Minus 2 Outside Function; Shift Down 2. P (2,5). Reflect Over X-Axis or Y-Axis 3. Stretches and compressions in both the vertical and horizontal planes. To the left zooms in, to the right zooms out. This skill will be useful as we progress in our study of mathematics. Download free on iTunes. Compare the graph of \(g\) and \(h\) to the basic squaring function defined by \(f(x)=x^{2}\), shown dashed in grey below: The function \(g\) is steeper than the basic squaring function and its graph appears to have been stretched vertically. WebTransformations of Functions DESMOS Online Calculator Exploration Activity. Answer: Figure 2.5.3. Make sure you are happy with the following topics before continuing. Develop some rules for this situation and share them on the discussion board. Mathway. We solve differential equations using the Laplace transform. Eg plot y = x 2 and y = x 2 + 3 on the same diagram, compare the graphs. WebThe Transformations Calculator is an online tool that finds the Laplace Transform of any given function in the time domain and presents the result in the frequency domain. Log InorSign Up. Get started with the video on the right, then dive deeper with the resources below. WebIn this video, I show you how to transform graphs using the list feature on the TI-84 Plus CE graphing calculator. The diagram shows the graph of y=f (x) y = f (x) and a point on the graph P (2,5). Determine whether the transformation is a translation or reflection. WebFunction Grapher is a full featured Graphing Utility that supports graphing up to 5 functions together. The order in which we apply horizontal and vertical translations does not affect the final graph. As we know, the Laplace transform of sin at = a/(s^2 + a^2). Get started with the video on the right, then dive deeper with the resources below. get Go. This is it. If a positive constant is added to the value in the domain before the function is applied, \(f(x + h)\), the graph will shift to the left. WebGraphing Calculator - MathPapa Graphing Calculator What do you want to calculate? Find out what the parent function is. Returns the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer. Step 2: Now click the button Submit to get the result. Two different options are possible: Perform a translation. Each transformation has the same effect on all functions. y = f (x). When the graph of a function is changed in appearance and/or location we call it a transformation. 58A set of operations that change the size and/or shape of a graph in a coordinate plane. Many teachers teach trig transformations without using t-charts; here is how you might do that for sin and cosine:. Here is a picture of the graph of g(x) =(0.5x)3+1. Parent function: For the two values of that are negative ( 2 and 1 ), replace the s with the from the absolute value ( 2 and 1, respectively) for those points. Functions can get pretty complex and go through transformations, like reflections along the x- or y-axis, shifts, stretching and shrinking, making the usual graphing techniques difficult. Use the points \(\{(1, 2), (0, 0), (1, 2)\}\) to graph the reflected and dilated function \(y=2|x|\). i.e., we now know the old x and y coordinates. Solve Now. A transformation is a method of altering a shapes size or position. It is necessary to comprehend not only the tables but also the formula in order to grasp the Laplace transform. Usage To plot a function just type it into the function box. Use rectangular, polar, cylindrical, or spherical coordinates. The following are the transformation matrices: In mathematics, a transformation is the transformation of one function into another function that may or may not be in the same domain. If you are one of You would insert it into the right hand part of the equation to get 2 2 = 4. Secondary full-load current, I 2 = (50 1000 / 400) = 125 A. b) Turns Ratio = N 1 / N 2 = V 1 / V 2 = (4000 / 400) = 10. Sketch the graph of \(g ( x ) = - ( x + 5 ) ^ { 2 } + 3\). 4.9. 582+ Math Consultants 13 Years in business Mathway. Look at the image above and ask your teacher which ones youre in charge of. Vertical and Horizontal Stretches/Compressions, 5. WebFunction Transformation Calculator How to graph your problem. WebGraph inequalities, contour plots, density plots and vector fields. A rigid transformation57 changes the location of the function in a coordinate plane, but leaves the size and shape of the graph unchanged. Use different colors to graph the family of graphs defined by \(y=kx^{2}\), where \(k \in \left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } \right\}\). WebFree calculator for transforming functions How to transform the graph of a function? \(y = \sqrt{x}\); Shift down \(5\) units; domain: \([0, )\); range: \([5, )\), 15. I go over how to enter numbers in lists. Graph \(f ( x ) = \sqrt { x }\) and \(g ( x ) = - \sqrt { x }\) on the same set of coordinate axes. However, the Laplace transform reduces a linear differential equation (LDE) to an algebraic equation, which may then be solved using basic algebraic identities. Step 3: Finally, the graph of the quadratic function will be displayed in the new window. If there is a whole number coefficient outside of parentheses, multiply the y-values of all points by that coefficient to see the graph expand vertically, for example, f(x) = 4x is stretched vertically by a factor of four. For example, lets say you had the function f: x x 2 and you wanted to transform (map) the number 2. Step 2: Multiply each x -value by 1 2. Eg plot y = x 2 and y = x 2 + 3 on the same diagram, compare the graphs. Download free in Windows Store. WebGraph inequalities, contour plots, density plots and vector fields. Solution. To put it another way, a Laplace transformation is just a quick way to solve a differential equation. WebGet the free "Reflection Calculator MyALevelMathsTutor" widget for your website, blog, Wordpress, Blogger, or iGoogle. Step 3: Finally, the Laplace transform of the given function will be displayed in the new window. WebLaplace Transform Formula: The standard form of unilateral laplace transform equation L is: F ( s) = L ( f ( t)) = 0 e s t f ( t) d t. Where f (t) is defined as all real numbers t 0 and (s) is a complex number frequency parameter. How do you find the transformation of a function? Download free on Google Play. by. 2. powered by. WebGraph Transformations. Since we can get the new period of the graph (how long it goes before repeating itself), by using \(\displaystyle \frac{2\pi }{b}\), and we know the phase shift, we can graph Loading Graph Transformations. Compare y = x 2 and y = x 2 + k where k is any integer, positive or negative. 2. powered by. 5.0. WebGet the free "Reflection Calculator MyALevelMathsTutor" widget for your website, blog, Wordpress, Blogger, or iGoogle. In this section, we will consider reflections about the \(x\)- and \(y\)-axis. Summary of Transformations To graph Draw the graph of f and: Changes in the equation of y = f(x) Vertical Shifts y = f (x) + c Visit Mathway on the web. Transformation of Points Vertical and Horizontal Stretches/Compressions 5. f(t) = L-1F(s). There are two types of transformations. Answer: Figure 2.5.3. There is a two-sided variation with the integral changing from to. In other words, if c > 1, then the graph is compressed. This introductory activity is designed to help students develop a better understanding of the relationship between the factors of a quadratic function and the x-intercepts of its related graph. \(y = \sqrt{x}\); Shift right \(2\) units and up \(1\) unit; domain: \([2, )\); range: \([1, )\), 17. The linearity equation can be used to answer the first step of the equation: (Using Linearity property of the Laplace transform), L(y)(s-2) + 5 = 1/(s-3) (Use value of y(0) ie -5 (given)), here (-5s+16)/(s-2)(s-3) can be written as -6/s-2 + 1/(s-3) using partial fraction method. Also, if a number is being added or subtracted inside parentheses, shift it left by that amount; if a number is being subtracted, move it right (opposite of what youd anticipate). Solution. Meanwhile, the application of the Laplace transformation is the best approach to turn differential equations into algebraic equations. \(h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } + 2 } & { \text { if } x < 0 } \\ { x + 2 } & { \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 3 \text { if } x < 0 } \\ { \sqrt { x } - 3 \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { l l } { x ^ { 3 } - 1 } & { \text { if } x < 0 } \\ { | x - 3 | - 4 } & { \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { c c } { x ^ { 3 } } & { \text { if } x < 0 } \\ { ( x - 1 ) ^ { 2 } - 1 } & { \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 1 } & { \text { if } x < 0 } \\ { 2 } & { \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { l l } { x + 2 } & { \text { if } x < 0 } \\ { ( x - 2 ) ^ { 2 } } & { \text { if } x \geq 0 } \end{array} \right.\), \(h ( x ) = \left\{ \begin{array} { l l } { ( x + 10 ) ^ { 2 } - 4 } & { \text { if } x < - 8 } \\ { x + 4 } & { \text { if } - 8 \leq x < - 4 } \\ { \sqrt { x + 4 } } & { \text { if } x \geq - 4 } \end{array} \right.\), \(f ( x ) = \left\{ \begin{array} { l l } { x + 10 } & { \text { if } x \leq - 10 } \\ { | x - 5 | - 15 } & { \text { if } - 10 < x \leq 20 } \\ { 10 } & { \text { if } x > 20 } \end{array} \right.\), \(h ( x ) = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 }\), \(h ( x ) = \frac { 1 } { 3 } ( x + 2 ) ^ { 2 }\), \(g ( x ) = - \frac { 1 } { 2 } \sqrt { x - 3 }\), \(g ( x ) = - \frac { 1 } { 4 } ( x + 3 ) ^ { 3 } - 1\), \(f ( x ) = - \frac { 1 } { x + 1 } + 2\).

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