Question

In Exercises 13–20, find the inverse of the function. Then graph the function and its inverse. $$ f(x)=-2 x+5 $$

Short answer

The inverse of the function \( f(x)=-2x+5 \) is \( y = -\frac{1}{2}x + \frac{5}{2} \).

Step-by-step solution

Find the inverse

To find the inverse of the function, first you swap \( x \) and \( y \). So, starting with \( y = -2x + 5 \), you swap \( x \) and \( y \) to get \( x = -2y + 5 \). Now, you rearrange this to get \( y \) on its own. This gives us the inverse function.

Rearrange to solve for y

You add \( 2y \) on both sides of the equation to isolate \( y \) on one side. That gives: \( 2y = -x + 5 \). Divide the whole equation by 2 to solve for y: \( y = -\frac{1}{2}x + \frac{5}{2} \) which is the inverse function.

Graph the function and its inverse

To plot the graphs of both functions, it is useful to note a few key points. For the original function \( f(x) = -2x + 5 \), the y-intercept is 5 and the slope is -2. For the inverse function, the y-intercept is \( \frac{5}{2} \) and the slope is \( -\frac{1}{2} \). The graph of the original function will be a line that decreases as \( x \) increases, and the graph of the inverse will be a line that also decreases but at a lesser slope.

Key concepts

Function Graphing

Graphing a function and its inverse involves plotting two distinct but related lines on a coordinate plane. These lines reflect off the line where the original function and its inverse meet at a symmetrical angle. To graph both functions:

• Identify key points such as y-intercepts and another point determined by the slope.
• For the function \( f(x) = -2x + 5 \):
• Y-intercept is 5 (where the line crosses the y-axis).
• Slope is -2, indicating the line descends steeply as \( x \) increases.
• For its inverse \( y = -\frac{1}{2}x + \frac{5}{2} \):
• Y-intercept is \( \frac{5}{2} \) (2.5 on the graph).
• Slope is \(-\frac{1}{2}\), showing a gentler descent.

When graphing, both lines should intersect at the line \( y = x \), which acts as a mirror between the original function and its inverse. This symmetry helps highlight the relationship between a function and its inverse.

Linear Functions

Linear functions are algebraic expressions that produce straight lines when plotted on a graph. They are characterized by their constant rate of change, known as the slope, and a starting point at the y-axis called the y-intercept. For instance, the function \( f(x) = -2x + 5 \) can be broken down into:

• Slope (-2): This value describes how the function changes as \( x \) changes. Here, the function decreases by 2 units for every 1 unit increase in \( x \).
• Y-intercept (5): At \( x = 0 \), the function value is 5. This is where the line crosses the y-axis.

Understanding these components helps us quickly sketch or visualize the graph of a linear function. The algebraic form \( y = mx + b \) is the general form for any linear function, where \( m \) represents the slope and \( b \) the y-intercept. Knowing how to identify and use these parts lets you graph any linear function easily.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to find the desired variable. This process is crucial when finding inverse functions. To find the inverse of \( f(x) = -2x + 5 \):

• Replace \( f(x) \) with \( y \). You start with \( y = -2x + 5 \).
• Swap \( x \) and \( y \). This gives \( x = -2y + 5 \).
• Isolate \( y \) by adding \( 2y \) to both sides, resulting in \( 2y = -x + 5 \).
• Divide each term by 2 to solve for \( y \). This gives \( y = -\frac{1}{2}x + \frac{5}{2} \), representing the inverse function.

Switching variables and simplifying aims to express the inverse function in the usual form \( y = mx + b \). Mastering these techniques allows you to tackle more complex algebraic tasks by breaking them into simple, logical steps.