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Chapter 1: Problem 18

For each function below, find \(f^{-1}(x)\). $$ f(x)=9+10 x $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{x - 9}{10} \).

Step by step solution

01

Understand the Function

The given function is a linear function, which is expressed as \( f(x) = 9 + 10x \). We will find its inverse, \( f^{-1}(x) \), by reversing the operations applied to \( x \). This involves undoing the addition and then the multiplication by 10.
02

Replace f(x) with y

First, replace \( f(x) \) with \( y \) for simplicity: \( y = 9 + 10x \). The goal is to express \( x \) in terms of \( y \).
03

Solve for x

To find \( x \), start by isolating it on one side of the equation. Subtract 9 from both sides: \[ y - 9 = 10x \] Then, divide both sides by 10 to solve for \( x \): \[ x = \frac{y - 9}{10} \]
04

Write the Inverse Function

Since \( y \) was just a placeholder for \( f(x) \), replace \( y \) with \( x \) in the equation from the previous step. The inverse function is: \( f^{-1}(x) = \frac{x - 9}{10} \).

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Key Concepts

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

Linear Functions
Linear functions are one of the simplest types of functions in mathematics. They have the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants.
  • \( m \) is the slope of the line, representing the rate of change.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
The line described by a linear function is straight.
This makes them easy to predict and analyze. For example, in the function \( f(x) = 9 + 10x \):
  • The slope \( m \) is 10, indicating a steep increase.
  • The y-intercept \( b \) is 9, meaning the line crosses at (0, 9) on the y-axis.
Linear functions are fundamental in understanding more complex relationships, making them a crucial part of algebra.
Solving Equations
Solving equations is a method to find the value of the variable that makes the equation true. When working with linear equations, the process typically involves:
  • Isolating the variable on one side of the equation.
  • Performing inverse operations to simplify the equation.
In the exercise with \( y = 9 + 10x \), our goal is to isolate \( x \). By following these steps:
Subtract 9 from both sides:
\( y - 9 = 10x \)
Next, divide everything by 10:
\( x = \frac{y - 9}{10} \)
This gives us \( x \) isolated, and thus we have solved for \( x \).
Knowing how to solve equations is key to finding the inverse of functions, as it allows us to express one variable in terms of another.
Function Operations
Function operations include several different types of processes that manipulate functions in various ways. Finding an inverse function is one of these essential operations.
When you find an inverse of a function, you essentially swap the input and output values.
Consider the original function \( f(x) = 9 + 10x \) and its inverse:
  • The inverse was found as \( f^{-1}(x) = \frac{x - 9}{10} \).
  • This means, for any output \( y \) of the original function, if you plug \( y \) into the inverse function, you should get back the input \( x \) of the original function.
This swap of input and output is central to understanding function inverses. It allows you to reverse processes, important in real-world applications where backward calculations are needed.

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Most popular questions from this chapter

Describe how each function is a transformation of the original function \(f(x)\). $$ f(x+4)-1 $$

Find the domain of each function. $$ f(x)=\frac{5 x+3}{4 x-1} $$

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ h(x)=-2|x-4|+3 $$

Sketch a graph of each piecewise function. $$ f(x)=\left\\{\begin{array}{ccc} |x| & \text { if } & x<2 \\ 5 & \text { if } & x \geq 2 \end{array}\right. $$

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ m(x)=\frac{1}{2} x^{3} $$
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