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Chapter 10: Problem 39

In the following exercises, find the inverse of each function. $$ f(x)=x-12 $$

Short Answer

Expert verified
The inverse function is f^{-1}(x) = x + 12.

Step by step solution

01

Replace f(x) with y

First, write the function using y instead of f(x). Therefore, the function becomes: y = x - 12
02

Swap x and y

To find the inverse function, swap the variables x and y. This gives us: x = y - 12
03

Solve for y

Next, isolate y by adding 12 to both sides of the equation: x + 12 = y Thus, the inverse function is: y = x + 12
04

Replace y with f^{-1}(x)

Finally, replace y with the notation for the inverse function, f^{-1}(x). Therefore, the inverse function is: 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.

Finding Inverses
Finding the inverse of a function essentially means determining a new function that reverses the effect of the original function. To identify an inverse function, follow these steps:
  • Replace the function notation, such as f(x), with y.
  • Swap the variables x and y in the equation.
  • Solve the resulting equation for y.
  • Replace y with the inverse notation f^{-1}(x) to indicate it's the inverse function.
For example, if you have the function f(x) = x - 12, you start by writing it as y = x - 12. Then, swap the variables to get x = y - 12 and solve for y by adding 12 to both sides, obtaining y = x + 12. Finally, write the inverse function as f^{-1}(x) = x + 12. Breaking this down makes it easier to understand that the inverse function 'undoes' the action of the original function.
Function Notation
Function notation, like f(x), is a concise way to represent functions. The notation indicates a function named f where x is the input variable. When converting an equation into a function notation, you simply follow the same relationship described by the equation. Here are a few key points about function notation:
  • f(x) is read as 'f of x' and represents the output of the function when x is the input.
  • To find the inverse of a function, use inverse notation such as f^{-1}(x).
  • The notation helps keep track of inputs and outputs, simplifying complex problems.
In our example, we started with f(x) = x - 12. After finding the inverse, we used f^{-1}(x) = x + 12 to represent the inverse function. This makes it clear and easy to understand which function we're working with at any step.
Solving Equations
Solving equations is a fundamental skill when working with functions and their inverses. Here's how to solve an equation when finding an inverse function:
  • First, rewrite the function as an equation using different variables if necessary (e.g., y = x - 12).
  • Next, swap the variables to set up the equation to solve for the new variable.
  • Solve for the isolated variable by performing algebraic operations like addition, subtraction, multiplication, or division.
To demonstrate, take the function y = x - 12. After swapping the variables to get x = y - 12, you solve for y by adding 12 to both sides, giving y = x + 12. Mastering these steps ensures you can effectively find inverse functions and understand the relationships between the original and inverse functions.

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Most popular questions from this chapter

In the following exercises, solve for \(x\). \(3 \log _{3} x=\log _{3} 27\)

Explain the method you would use to solve these equations: \(3^{x+1}=81, \quad 3^{x+1}=75 .\) Does your method require logarithms for both equations? Why or why not?

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(4^{x}=112\)

Use an example to show that $$ \log (a+b) \neq \log a+\log b $$
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