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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 49abc
Chapter 3, Problem 49abc

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = √(x-1)

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Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = √(x - 1). Next, swap x and y to reflect the inverse relationship. This results in x = √(y - 1).
Step 2: Solve for y in terms of x to find the inverse function. Square both sides of the equation to eliminate the square root: x² = y - 1. Then, isolate y by adding 1 to both sides: y = x² + 1. Thus, the inverse function is ƒ¯¹(x) = x² + 1.
Step 3: To graph both f(x) and ƒ¯¹(x), plot f(x) = √(x - 1), which is a square root function starting at x = 1 and increasing to the right. For ƒ¯¹(x) = x² + 1, plot a parabola that opens upwards with its vertex at (0, 1). Ensure both graphs are in the same rectangular coordinate system.
Step 4: Determine the domain and range of f(x). Since f(x) = √(x - 1), the domain is [1, ∞) because the square root function is only defined for x - 1 ≥ 0. The range is [0, ∞) because the square root function outputs non-negative values.
Step 5: Determine the domain and range of ƒ¯¹(x). Since ƒ¯¹(x) = x² + 1, the domain is [0, ∞) because the input to the inverse function corresponds to the range of f(x). The range of ƒ¯¹(x) is [1, ∞) because the output of the inverse function corresponds to the domain of f(x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function reverses the effect of the original function. For a function f(x), its inverse f¯¹(x) satisfies the condition f(f¯¹(x)) = x for all x in the domain of f¯¹. To find the inverse, one typically swaps the roles of x and y in the equation and solves for y.
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Graphing Logarithmic Functions

Domain and Range

The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. Understanding the domain and range is crucial for both the original function and its inverse, as they are interchanged in inverse functions.
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Domain & Range of Transformed Functions

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input and output values. When graphing both a function and its inverse, the graphs will be symmetric with respect to the line y = x, illustrating how each function undoes the other.
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Related Practice
Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

Textbook Question

Graph each equation in a rectangular coordinate system. y = -2

Textbook Question

In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. x² + (y − 1)² = 1

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Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

Textbook Question

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)