Math Problem Statement

The function f(x) = x + 8 is one-to-one. a. Find an equation for f⁻¹(x), the inverse function. b. Verify that your equation is correct by showing that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Select the correct choice below.

Solution

Let's analyze the problem step by step.

Given Function

The function is: f(x)=x+8f(x) = x + 8 We are tasked to find the inverse function f1(x)f^{-1}(x).


Step 1: Find the Inverse f1(x)f^{-1}(x)

To find the inverse function:

  1. Start by replacing f(x)f(x) with yy: y=x+8y = x + 8

  2. Solve for xx in terms of yy: x=y8x = y - 8

  3. Replace yy with xx to express the inverse: f1(x)=x8f^{-1}(x) = x - 8

Thus, the inverse function is: f1(x)=x8f^{-1}(x) = x - 8


Step 2: Verify the Inverse

To verify f1(x)=x8f^{-1}(x) = x - 8:

  1. Check f(f1(x))f(f^{-1}(x)): f(f1(x))=f(x8)=(x8)+8=xf(f^{-1}(x)) = f(x - 8) = (x - 8) + 8 = x

  2. Check f1(f(x))f^{-1}(f(x)): f1(f(x))=f1(x+8)=(x+8)8=xf^{-1}(f(x)) = f^{-1}(x + 8) = (x + 8) - 8 = x

Both conditions are satisfied, so f1(x)=x8f^{-1}(x) = x - 8 is correct.


Step 3: Select the Correct Choice

Looking at the answer options:

  • Option B matches the correct inverse function f1(x)=x8f^{-1}(x) = x - 8, where the condition x0x \neq 0 is irrelevant to the general solution.

Thus, Option B is the correct choice.


Final Answer:

The correct choice is B: f1(x)=x8f^{-1}(x) = x - 8


Would you like more details on how to verify the inverse, or any clarification? Here are five related questions for practice:

  1. How do you determine if a function is one-to-one?
  2. What is the general process to find the inverse of a linear function?
  3. How do you verify if two functions are inverses of each other?
  4. Can a function have more than one inverse? Why or why not?
  5. How does the graph of a function compare to its inverse?

Tip: To find an inverse, swap xx and yy, then solve for yy.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Linear Equations

Formulas

f(x) = x + 8
f⁻¹(x) = x - 8

Theorems

Verification of Inverse Functions

Suitable Grade Level

Grade 9-11