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Chapter 13: Problem 48

If \(f(x)=x-12,\) show that \(f^{-1}(x)=x+12\)

Short Answer

Expert verified
To show that the inverse of \(f(x) = x - 12\) is \(f^{-1}(x) = x + 12\), we replace \(f(x)\) with \(y\), swap \(x\) and \(y\), and solve for \(y\). Doing so, we find that the inverse function is indeed \(f^{-1}(x) = x + 12\).

Step by step solution

01

Replace f(x) with y

Write the given function with y instead of f(x): \[y = x - 12\]
02

Swap x and y

Swap x and y in the equation: \[x = y - 12\]
03

Solve for y

To find the inverse function, solve the equation for y: Add 12 to both sides of the equation: \[x + 12 = y\] Now, we have the inverse function: \[f^{-1}(x) = x + 12\] Thus, the inverse of the given function f(x) = x - 12 is indeed f^(-1)(x) = x + 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Functions
An inverse function essentially undoes what the original function does. Think of it as a reverse operation. If a function maps input to output, its inverse will map that output back to the input. This is crucial because it allows us to retrace our steps in mathematical operations.
  • For the function \( f(x) = x - 12 \), think of it as taking a number, removing 12, and receiving a new number.
  • The inverse \( f^{-1}(x) = x + 12 \) indicates that we now take that result and add 12 back to return to the original number.
Understanding inverses helps us make sense of how numbers and operations relate to each other. Not all functions have inverses, only bijective (one-to-one and onto) ones do. This means each output must be generated from one unique input, ensuring there are no duplicates. Calculating inverse functions is a foundational skill in algebra and calculus, enhancing problem-solving capabilities.
Function Manipulation
Manipulating functions involves altering and solving equations to reveal different properties or to find inverses like we did in the exercise. The first step often involves changing the function's form.
  • We first replaced \( f(x) \) with \( y \) to make the equation easier to work with: \( y = x - 12 \).
  • The critical next step was to swap the roles of \( x \) and \( y \): we wrote \( x = y - 12 \). This step conceptually puts us in the frame of solving for the input when given the output.
Function manipulation ensures we can adjust and rearrange equations to solve them efficiently. It is a key skill not only in algebra but in many mathematical applications, such as calculus and differential equations, where altering functions provides deeper insights into their behavior and relationships.
Algebraic Equations
Solving algebraic equations is a fundamental skill in finding inverse functions. It involves isolating a variable to express it solely in terms of other variables or constants. This approach was used in the exercise to identify the inverse function.
First, after the function \( x = y - 12 \) was established, we needed to solve for \( y \):
  • This required us to add 12 to both sides, transforming it into \( x + 12 = y \).
  • Rewriting the equation in this way gave us the inverse function: \( f^{-1}(x) = x + 12 \).
Algebraic manipulation is a critical process, and understanding how to manipulate equations is essential for solving numerous mathematical problems. It allows us to transition fluidly between different forms of expressions, and is the backbone of solving equations in both academic and practical instances.

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