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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 41abc
Chapter 3, Problem 41abc

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 of f and ƒ¯¹. ƒ(x) = x² − 4, x ≥ 0

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Step 1: To find the inverse function ƒ¯¹(x), start by replacing ƒ(x) with y. So, rewrite the given function as y = x² - 4, where x ≥ 0. Then, swap x and y to begin solving for y. This gives x = y² - 4.
Step 2: Solve for y in terms of x. Add 4 to both sides of the equation to isolate the y² term: x + 4 = y². Then, take the square root of both sides to solve for y. Since x ≥ 0, we only consider the positive square root: y = √(x + 4). Thus, the inverse function is ƒ¯¹(x) = √(x + 4).
Step 3: To graph ƒ(x) and ƒ¯¹(x) on the same coordinate system, plot the parabola ƒ(x) = x² - 4 for x ≥ 0 (this is the right half of the parabola). Then, plot the square root function ƒ¯¹(x) = √(x + 4), which is the reflection of ƒ(x) across the line y = x.
Step 4: Determine the domain and range of ƒ(x). Since ƒ(x) = x² - 4 and x ≥ 0, the domain of ƒ(x) is [0, ∞). The range of ƒ(x) is [-4, ∞) because the smallest value of ƒ(x) occurs when x = 0, giving ƒ(0) = -4.
Step 5: Determine the domain and range of ƒ¯¹(x). The domain of ƒ¯¹(x) is the range of ƒ(x), which is [-4, ∞). The range of ƒ¯¹(x) is the domain of ƒ(x), which is [0, ∞). Use interval notation to express these relationships clearly.

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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. Understanding this concept is crucial for finding the equation of f¯¹(x) in the given problem.
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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. For the function f(x) = x² - 4 with the restriction x ≥ 0, the domain is [0, ∞) and the range is [-4, ∞). Knowing how to determine the domain and range is essential for part c of the question.
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Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input and output values. For the function f(x) = x² - 4, the graph is a parabola opening upwards, and its inverse will reflect across the line y = x. Understanding how to graph both f and its inverse is necessary for part b of the question, as it helps visualize their relationship.
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