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: We are tasked to find the inverse function .
Step 1: Find the Inverse
To find the inverse function:
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Start by replacing with :
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Solve for in terms of :
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Replace with to express the inverse:
Thus, the inverse function is:
Step 2: Verify the Inverse
To verify :
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Check :
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Check :
Both conditions are satisfied, so is correct.
Step 3: Select the Correct Choice
Looking at the answer options:
- Option B matches the correct inverse function , where the condition is irrelevant to the general solution.
Thus, Option B is the correct choice.
Final Answer:
The correct choice is B:
Would you like more details on how to verify the inverse, or any clarification? Here are five related questions for practice:
- How do you determine if a function is one-to-one?
- What is the general process to find the inverse of a linear function?
- How do you verify if two functions are inverses of each other?
- Can a function have more than one inverse? Why or why not?
- How does the graph of a function compare to its inverse?
Tip: To find an inverse, swap and , then solve for .
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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