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Chapter 10: Problem 40

In the following exercises, find the inverse of each function. $$ f(x)=x+17 $$

Short Answer

Expert verified
\( f^{-1}(x) = x - 17 \)

Step by step solution

01

- Understand the Function

Identify the given function: \( f(x) = x + 17 \). This is a linear function, where the output is achieved by adding 17 to the input, \( x \).
02

- Replace f(x) with y

To find the inverse, first rewrite the function \( f(x) \) using \( y \) instead: \( y = x + 17 \). This will help simplify the next steps.
03

- Swap Variables

Interchange \( x \) and \( y \) to begin solving for the inverse function: \( x = y + 17 \). Now, solve this new equation for \( y \).
04

- Isolate y

Isolate \( y \) in the equation \( x = y + 17 \) by subtracting 17 from both sides: \( y = x - 17 \).
05

- Write the Inverse Function

Replace \( y \) with \( f^{-1}(x) \) to express the inverse function properly: \( f^{-1}(x) = x - 17 \).

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Key Concepts

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

linear function
Linear functions are fundamental in algebra. A linear function can be described by the equation:y = mx + bwhere **m** is the slope, and **b** is the y-intercept. In simple terms, the slope **m** indicates how steep the line is, and the y-intercept **b** tells us where the line crosses the y-axis. For example, in the functionf(x) = x + 17this is a linear function where the slopem = 1and the y-intercept b = 17. All linear functions graph as straight lines and have unique properties. They grow constantly. In our exercise, the linear function grows with each step by adding 17.
solving equations
Solving equations can feel like solving a puzzle, where the goal is to find the value of the variable that makes the equation true. Here are the basic steps:
  • Identify the equation to solve
  • Isolate the variable on one side
  • Perform the same operations on both sides of the equation
For our functiony = x + 17, we follow these steps to solve for y. If we want to isolate y, we must perform the inverse operation. Since 17 is added to x, we need to subtract 17 from both sides to isolatey: x - 17 = y. This operation helps us find the inverse function.
function notation
Function notation is a way to clearly define functions in algebra. When we write a function asf(x), it means that fdepends on x. It's often read as 'f of x.' This notation succinctly shows the relationship between the input xand the output f(x). For example, iff(x) = x + 17, thenf(3) = 3 + 17 = 20. The function notation helps when we are working with inverses too. The inverse function is denoted asf^{-1}(x), which reverses the original function's operations. To find the inverse, replace f(x) with y, interchange x and y, solve for y, and then renote yasf^{-1}(x). Following these steps forf(x) = x + 17, we get an inversef^{-1}(x) = x - 17.

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Most popular questions from this chapter

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{3}\left(\sqrt{2} x^{2}\right)\)

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, solve each equation. \(\log _{2}(x+2)-\log _{2}(2 x+9)=-\log _{2} x\)

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm. \(\log _{15} 93\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\)
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