Textbook Question
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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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 10
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 function f(-x) reflects the graph of f(x) across the y-axis. This means the x-coordinates of the points on the graph will be negated.
Step 3: Apply the reflection transformation f(-x). The original points (1, -3) and (4, -3) will become (-1, -3) and (-4, -3), respectively. The horizontal line segment is now between (-4, -3) and (-1, -3).
Step 4: Apply the vertical shift of +3 to the reflected graph. This means adding 3 to the y-coordinates of all points on the graph. The points (-4, -3) and (-1, -3) will become (-4, 0) and (-1, 0), respectively.
Step 5: Plot the transformed graph g(x). The new graph is a horizontal line segment from (-4, 0) to (-1, 0).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformations involve altering the graph of a function through shifts, stretches, or reflections. In this case, the function g(x) = f(-x) + 3 represents a horizontal reflection of f(x) across the y-axis, followed by a vertical shift upward by 3 units. Understanding these transformations is crucial for accurately graphing the new function.
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Domain & Range of Transformed Functions
Reflection Across the Y-Axis
Reflecting a function across the y-axis means that for every point (x, y) on the original graph, there is a corresponding point (-x, y) on the reflected graph. This transformation changes the sign of the x-coordinates while keeping the y-coordinates the same, which is essential for graphing g(x) = f(-x).
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5:00Reflections of Functions
Vertical Shift
A vertical shift involves moving the entire graph of a function up or down without altering its shape. In the function g(x) = f(-x) + 3, the '+3' indicates that the graph of f(-x) will be shifted upward by 3 units. This shift affects the y-coordinates of all points on the graph, which is important for determining the final position of g(x).
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5:34Shifts of Functions
Related Practice
Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = -f(x) +3
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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 − 4) and g(x) = x³ +4
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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
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Textbook Question
Find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)
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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 (−2, 2) and parallel to the line whose equation is 2x-3y-7=0
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