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Chapter 12: Problem 22

Determine whether each function is one-to-one. If it is, find the inverse. $$ g(x)=-6 x-8 $$

Short Answer

Expert verified
The function is one-to-one. The inverse is \( g^{-1}(x) = -\frac{x + 8}{6} \).

Step by step solution

01

Determine if the function is one-to-one

A function is one-to-one if every y-value has a unique x-value. For linear functions of the form \( y = mx + b \), where m is the slope, the function is one-to-one if m \eq 0. In this case, the function is \( g(x) = -6x - 8 \). The slope, m, is -6, which is not equal to 0. Therefore, \( g(x) \) is one-to-one.
02

Find the inverse of the function

To find the inverse, swap the variables x and y and solve for y. Start with \( y = -6x - 8 \). Swap x and y to get \( x = -6y - 8 \). Then solve for y:1. Add 8 to both sides: \( x + 8 = -6y \)2. Divide both sides by -6: \( y = -\frac{x + 8}{6} \)Thus, the inverse function is \( g^{-1}(x) = -\frac{x + 8}{6} \).

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Key Concepts

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

Inverse Function
An inverse function essentially reverses the roles of the input and output. If you have a function \( g(x) = -6x - 8 \), its inverse would switch \( x \) and \( y \) in the equation. By solving for \( y \), you find the inverse function, denoted as \( g^{-1}(x) \). Here's a simple way to think about it: if a function pairs \( x \) with \( y \), its inverse will pair \( y \) with \( x \). For the function \( g(x) = -6x - 8 \):
  • Swap \( x \) and \( y \): \( x = -6y - 8 \).
  • Solve for \( y \): Add 8 to both sides giving \( x + 8 = -6y \).
  • Finally, divide both sides by -6: \( y = -\frac{x + 8}{6} \).
The inverse function here is \( g^{-1}(x) = -\frac{x + 8}{6} \). An important thing to remember is that a function must be one-to-one to have an inverse. This leads us to our next concept, linear functions, and how we determine if they are one-to-one.
Linear Functions
Linear functions are of the form \( y = mx + b \), where \( m \) is called the slope and \( b \) is the y-intercept. The defining feature of a linear function is that it graphs as a straight line.
To determine whether a linear function is one-to-one, look at its slope (\( m \)):
  • If \( m eq 0 \), the function is one-to-one because each \( y \) value has a unique \( x \) value.
  • If \( m = 0 \), the function is a horizontal line, meaning multiple \( x \) values map to the same \( y \) value, so it is not one-to-one.
For the function \( g(x) = -6x - 8 \), the slope \( m = -6 \). Since -6 is not equal to 0, the function is one-to-one and we can find its inverse.
Linear functions are fundamental in understanding basic algebra and how changes in \( x \) affect \( y \). Let's move on to understanding the slope, which plays a crucial role in these functions.
Slope
The slope of a linear function \( y = mx + b \) is the coefficient \( m \). It measures the steepness of the line. Here's how slope affects the function:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • A slope of zero means the line is horizontal.
The slope is calculated by \( m = \frac{\Delta y}{\Delta x} \), which means the change in \( y \) over the change in \( x \). For the function \( g(x) = -6x - 8 \), the slope is \( -6 \). This tells us the line decreases by 6 units in \( y \) for every 1 unit increase in \( x \). The concept of slope is key to understanding how linear functions behave and is essential for identifying one-to-one functions in general.

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Most popular questions from this chapter

Solve each equation. \(\log _{1 / 3}(x-4)=2\)

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{6} \sqrt[3]{5}\)

$$ \left(\frac{4}{3}\right)^{x}=\frac{27}{64} $$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{\ln (6-x)}=e^{\ln (4+2 x)} $$

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=-4 x $$
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