Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 11
Use the graph of y = f(x) to graph each function g.
g(x) = ½ f(x)
Verified step by step guidance1
Step 1: Observe the graph of y = f(x). It is a horizontal line segment from (1, -3) to (4, -3). This means that the function f(x) has a constant value of -3 for all x in the interval [1, 4].
Step 2: The new function g(x) = ½ f(x) involves scaling the values of f(x) by a factor of ½. Since f(x) = -3, g(x) = ½ * (-3) = -3/2 for all x in the interval [1, 4].
Step 3: To graph g(x), keep the x-values the same (from 1 to 4), but adjust the y-values to reflect the scaled value of -3/2. This means the new graph will be a horizontal line segment at y = -3/2.
Step 4: Plot the points (1, -3/2) and (4, -3/2) on the graph. These points represent the endpoints of the horizontal line segment for g(x).
Step 5: Draw a horizontal line segment connecting the points (1, -3/2) and (4, -3/2). This is the graph of g(x) = ½ f(x).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the process of altering the graph of a function through operations such as scaling, translating, or reflecting. In this case, the function g(x) = ½ f(x) represents a vertical compression of the function f(x) by a factor of ½, meaning that all y-values of f(x) are halved, resulting in a graph that is closer to the x-axis.
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Domain & Range of Transformed Functions
Graphing Linear Functions
Graphing linear functions involves plotting points that satisfy the function's equation and connecting them to form a straight line. For the function f(x) shown in the graph, which is a horizontal line at y = -3 between x = 1 and x = 4, understanding how to plot these points is essential for accurately representing the transformed function g(x).
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5:26Graphs of Logarithmic Functions
Horizontal and Vertical Scaling
Horizontal and vertical scaling are techniques used to stretch or compress the graph of a function. Vertical scaling, as seen in g(x) = ½ f(x), compresses the graph vertically, affecting the y-values while keeping the x-values unchanged. This concept is crucial for predicting how the graph of g will appear compared to f.
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5:28Horizontal Parabolas
Related Practice
Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = 2f(x)
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Textbook Question
Find the domain of each function. f(x) = 1/(x2+1) - 1/(x2-1)
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Textbook Question
Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (4, −7) and perpendicular to the line whose equation is x − 2y – 3 = 0
Textbook Question
The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = x +3
Textbook Question
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3.5, 8.2) and (-0.5, 6.2)
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