Problem 17 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ g(x)=-2 x $$

Step-by-Step Solution

Answer

The inverse function is \( g^{-1}(x) = \frac{-1}{2}x \).

Step 1: Original Function Observation

The given function is a linear function, which is expressed as a simple linear equation: \( g(x) = -2x \).

Step 1: Replace \(g(x)\) with \(y\)

To make the process of finding the inverse easier, rewrite the function as \( y = -2x \).

Step 2: Swap Variables \(x\) and \(y\)

To find the inverse, we need to switch the roles of \(x\) and \(y\) in the equation. This gives us \( x = -2y \).

Step 3: Solve for \(y\)

Rearrange the equation \( x = -2y \) to solve for \(y\). This involves dividing both sides by \(-2\) to isolate \(y\): \[ y = \frac{-1}{2}x \].

Step 4: Express the Inverse Function

Now express the solution from the previous step as the inverse function. The inverse of \( g(x) = -2x \) is \( g^{-1}(x) = \frac{-1}{2}x \).

Step 5: Graph Both Functions

To graph the functions, plot both the original function \( g(x) = -2x \) and its inverse \( g^{-1}(x) = \frac{-1}{2}x \) on the same coordinate plane. The graph of the original function is a line with a slope of \(-2\) passing through the origin. The graph of the inverse function is a line with a slope of \(-\frac{1}{2}\) also passing through the origin.

Key Concepts

Understanding Linear Functions

Linear functions are often represented by the equation \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. These functions create straight lines when graphed. In the case of the function \( g(x) = -2x \), we can see it follows the form \( y = mx + c \) with \( m = -2 \) and \( c = 0 \). This means:

• The line has a slope of \(-2\).
• The line crosses the y-axis at the origin (0,0), as there's no constant \( c \) added.

Linear functions are fundamental as they model direct proportionality and appear frequently in various real-life contexts, like speed, cost calculations, etc. They are easy to work with due to their simplicity.

Exploring Graphing Functions

Graphing functions helps us visualize mathematical relationships. When graphing \( g(x) = -2x \), we begin by determining its slope and intercept, as discussed.

• The slope \(-2\) indicates the line falls two units vertically for each unit it moves horizontally.
• The line passes through (0,0) since the intercept is 0.

Here's how to plot it: Start at the origin, move down two units, and one unit right to find your next point. Repeat this pattern to sketch the line. When graphing inverse functions like \( g^{-1}(x) = \frac{-1}{2}x \), observe the reflection along the line \( y = x \), since the roles of \( x \) and \( y \) interchange, leading to a flatter slope and a similar starting point at the origin.

Step-by-Step Solving Equations

Solving equations to find an inverse involves swapping variables and solving for the new dependent variable. For the function \( y = -2x \):

• First, swap \( x \) and \( y \) to get \( x = -2y \).
• Next, solve for \( y \) by dividing both sides by \(-2\), yielding \( y = \frac{-1}{2}x \).

The expression \( y = \frac{-1}{2}x \) is the inverse function \( g^{-1}(x) \), indicating a reversal of roles where previous inputs become outputs and vice versa. Mastering this process is crucial for understanding how changes in one variable affect another in inverse relationships. It's a core concept in algebra that extends to various applications in calculus and beyond.