Textbook Question
Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = = -x and g(x) = -x
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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 9
Use the graph of y = f(x) to graph each function g.
g(x) = -f(x) +3
Verified step by step guidance1
Step 1: Analyze the given graph of y = f(x). The graph is a horizontal line segment from (1, -3) to (4, -3). This means that f(x) = -3 for all x in the interval [1, 4].
Step 2: Understand the transformation g(x) = -f(x) + 3. The transformation involves two steps: (a) Negate the values of f(x), which means flipping the graph vertically, and (b) Add 3 to the result, which shifts the graph upward by 3 units.
Step 3: Apply the negation transformation. Since f(x) = -3, negating it gives -f(x) = 3. The graph is now a horizontal line segment at y = 3 for x in [1, 4].
Step 4: Apply the vertical shift transformation. Adding 3 to -f(x) gives g(x) = 3 + 3 = 6. The graph is now a horizontal line segment at y = 6 for x in [1, 4].
Step 5: Plot the transformed graph of g(x). The new graph is a horizontal line segment from (1, 6) to (4, 6).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the changes made to the graph of a function based on modifications to its equation. In this case, the function g(x) = -f(x) + 3 involves a vertical shift and a reflection. The negative sign indicates a reflection over the x-axis, while the '+3' shifts the graph upward by three units.
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Domain & Range of Transformed Functions
Reflection Over the X-Axis
Reflecting a function over the x-axis means that for every point (x, y) on the original graph, the corresponding point on the reflected graph will be (x, -y). This transformation changes the sign of the output values, effectively flipping the graph upside down. For the function g(x) = -f(x), this reflection alters the horizontal line segment of f(x) from y = -3 to y = 3.
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Vertical Shift
A vertical shift occurs when a constant is added to or subtracted from a function's output. In the function g(x) = -f(x) + 3, the '+3' indicates that the entire graph of -f(x) is moved up by three units. This means that after reflecting the original function, the new graph will be positioned higher on the y-axis, resulting in a horizontal line segment at y = 0.
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5:34Shifts of Functions
Related Practice
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Evaluate each function at the given values of the independent variable and simplify. (a) f(-2), (b) f(1), (c) f(2)
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Use the graph of y = f(x) to graph each function g.
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