Textbook Question
Find a. (fog) (x) b. (go f) (x) f(x)=4x-3, g(x) = 5x² - 2
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 55
In Exercises 55–59, use the graph of to graph each function g.
g(x) = f(x + 2) + 3
Verified step by step guidance1
Identify the base function f(x) from the graph. Here, f(x) is an absolute value function with vertex at (0,0), forming a V-shape.
Understand the transformation in g(x) = f(x + 2) + 3. The term (x + 2) inside the function indicates a horizontal shift, and the +3 outside indicates a vertical shift.
Apply the horizontal shift: Replace x with (x + 2) shifts the graph of f(x) to the left by 2 units. This moves the vertex from (0,0) to (-2,0).
Apply the vertical shift: Adding +3 outside the function shifts the graph up by 3 units. This moves the vertex from (-2,0) to (-2,3).
Sketch the new graph g(x) by shifting every point of f(x) left 2 units and up 3 units, maintaining the same shape and slopes of the V.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformations
Function transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this problem, the function g(x) = f(x + 2) + 3 represents a horizontal shift left by 2 units and a vertical shift up by 3 units of the original function f(x). Understanding these shifts helps in graphing the transformed function accurately.
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Domain & Range of Transformed Functions
Horizontal Shifts
A horizontal shift occurs when the input variable x is replaced by (x + h) or (x - h). For g(x) = f(x + 2), the graph of f(x) shifts left by 2 units because adding inside the function moves the graph in the opposite direction of the sign. This concept is crucial for repositioning the vertex of the V-shaped graph.
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Vertical Shifts
A vertical shift happens when a constant is added or subtracted outside the function, such as f(x) + k. In g(x) = f(x + 2) + 3, adding 3 shifts the entire graph of f(x + 2) upward by 3 units. This moves the vertex and all points on the graph vertically, maintaining the shape but changing the position.
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Related Practice
Textbook Question
In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. (go f) (-1)
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Use the vertical line test to identify graphs in which y is a function of x.
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Graph each equation in a rectangular coordinate system. f(x) = 1
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Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (x − 2)²