Problem 26 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ g(x)=\frac{2 x+3}{6} $$

Step-by-Step Solution

Answer

The inverse of the function is \( g^{-1}(x) = \frac{6x - 3}{2} \).

Step 1: Identify the Function

The given function is \( g(x) = \frac{2x + 3}{6} \). This is a linear function and can be written as \( y = \frac{2x + 3}{6} \), where \( y \) represents the output of \( g(x) \).

Step 2: Swap Variables

In order to find the inverse, we first write the function as \( y = \frac{2x + 3}{6} \). Then, we swap \( x \) and \( y \). This gives us the equation \( x = \frac{2y + 3}{6} \).

Step 3: Solve for the New Function (y)

Solve for \( y \): Our goal is to solve the equation \( x = \frac{2y + 3}{6} \) for \( y \). First, multiply both sides by 6 to get rid of the fraction: \( 6x = 2y + 3 \).

Step 4: Isolate y

Subtract 3 from both sides to isolate the term with \( y \): \( 6x - 3 = 2y \).

Step 5: Find the Inverse Function

Divide both sides by 2 to solve for \( y \): \( y = \frac{6x - 3}{2} \). Therefore, the inverse function is \( g^{-1}(x) = \frac{6x - 3}{2} \).

Step 6: Graphing the Functions

To graph, plot both the original function \( g(x) = \frac{2x + 3}{6} \) and the inverse \( g^{-1}(x) = \frac{6x - 3}{2} \). These graphs will reflect over the line \( y = x \).

Step 7: Verify Inverses

To confirm that the functions are inverses, check that \( g(g^{-1}(x)) = x \) and \( g^{-1}(g(x)) = x \). Substitute \( g^{-1}(x) \) into \( g(x) \) and \( g(x) \) into \( g^{-1}(x) \) to verify they both simplify to \( x \).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions, characterized by a straight line when graphed on the Cartesian plane. A linear function can be expressed in the form \( y = mx + b \), where:

• \( m \) is the slope, determining the steepness of the line.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In the context of our problem, the function \( g(x) = \frac{2x + 3}{6} \) is a linear function. Simplifying it, we get \( y = \frac{2}{6}x + \frac{3}{6} \), or \( y = \frac{1}{3}x + \frac{1}{2} \). This tells us that the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is \( \frac{1}{2} \).
Understanding these components can help in graphing the function and finding its inverse.

Graphing Functions

Graphing functions involves plotting points that satisfy the function's equation and then connecting them to reveal the shape of the function. With linear functions, this process is straightforward, as you primarily need two points to draw the line.
To graph \( g(x) = \frac{2x + 3}{6} \):

• Start by finding the y-intercept, \( (0, \frac{1}{2}) \), which is the value of \( y \) when \( x = 0 \).
• Use the slope \( \frac{1}{3} \) to find another point. Slope means rise over run, so starting from \( (0, \frac{1}{2}) \), move one unit up and three units to the right to locate the next point, \( (3, \frac{5}{2}) \).

Now that you have two points, you can draw the line through these points to complete your graph for \( g(x) \). The inverse function, \( g^{-1}(x) = \frac{6x - 3}{2} \), is found similarly by identifying key points through its slope and y-intercept, and is reflected over the line \( y = x \).

Inverse Function Verification

Verifying that two functions are inverses of each other involves checking whether applying one function after the other returns the original input value. For our functions, \( g(x) = \frac{2x + 3}{6} \) and its inverse \( g^{-1}(x) = \frac{6x - 3}{2} \), we follow these steps:

• Calculate \( g(g^{-1}(x)) \).
• Substitute \( g^{-1}(x) \) into \( g(x) \), simplifying to show that it equals \( x \).
• Next, perform the calculation \( g^{-1}(g(x)) \).
• Substitute \( g(x) \) into \( g^{-1}(x) \), and simplify, aiming to also confirm that it equals \( x \).

If both substitutions simplify to \( x \) as expected, then the inverse relationship is verified, completing the process of confirming these functions are true inverses of each other. This step is essential to demonstrate the reciprocal nature between a function and its inverse.