Problem 48 - Intermediate Algebra

Question

(a) find the inverse of the given function, and (b) graph the given function and its inverse on the same set of axes. (Objective 4) $$f(x)=-3 x+9$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = -\frac{1}{3}x + 3 \). Graph both functions to see their symmetry around the line \( y=x \).

Step 1: Understand the Problem

We need to find the inverse of the function \( f(x) = -3x + 9 \) and then graph both the function and its inverse on a coordinate plane.

Step 2: Express the Function as an Equation

First, express \( f(x) = -3x + 9 \) as an equation: \( y = -3x + 9 \). This helps us when we solve for \( x \) to find the inverse.

Step 3: Swap Variables and Solve for x

To find the inverse, swap \( x \) and \( y \) in the equation: \( x = -3y + 9 \). Now, solve for \( y \) to get the inverse function.

Step 4: Isolate y

Rearrange the equation \( x = -3y + 9 \) to isolate \( y \):\[ x - 9 = -3y \] \[ y = \frac{x - 9}{-3} \] \[ y = -\frac{1}{3}x + 3 \]. Thus, the inverse function is \( f^{-1}(x) = -\frac{1}{3}x + 3 \).

Step 5: Verify the Inverse

To verify, check that applying \( f(x) \) and \( f^{-1}(x) \) successively yields \( x \): \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This ensures the correctness of the inverse.

Step 6: Graph the Functions

Graph \( f(x) = -3x + 9 \) and \( f^{-1}(x) = -\frac{1}{3}x + 3 \) on the same set of axes. Use suitable scales to mark the points and draw the lines. The original function will have a steep decline, while the inverse will rise gently.

Step 7: Analyze the Graph

Note that the graph of \( y = x \) is the line of symmetry between a function and its inverse. The point where the original function and its inverse intersect this line shows correct reflection.

Key Concepts

Graphing Functions

Graphing functions is a crucial skill in understanding how mathematical relationships behave visually. It helps you see the relationship between variables and how they change relative to each other. For instance, plotting the function \( f(x) = -3x + 9 \) involves determining key points. By substituting different \( x \) values into the function, you get corresponding \( y \) values:

• When \( x = 0 \), \( y = 9 \)
• When \( x = 3 \), \( y = 0 \)
• When \( x = -3 \), \( y = 18 \)

Connecting these points gives a line that represents the function. To graph the inverse, \( f^{-1}(x) = -\frac{1}{3}x + 3 \), follow a similar process.

• When \( x = 0 \), \( y = 3 \)
• When \( x = 9 \), \( y = 0 \)
• When \( x = -3 \), \( y = 4 \)

Plotting these points and connecting them shows the graph of the inverse function. Remember, both functions will reflect over the line \( y = x \). The intersection points and the way the lines cross the axes offer useful insights into the behaviors and properties of the functions.

Linear Functions

Linear functions are one of the simplest types of functions, but they hold significant power in mathematics. They have the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This equation represents a straight line, illustrating how changes in \( x \) affect \( y \) directly. Take \( f(x) = -3x + 9 \) as an example. Here:

• The slope \( m \) is \(-3\), indicating a decreasing line.
• The y-intercept \( b \) is \(9\), meaning the line crosses the y-axis at \(9\).

Understanding the slope and intercept allows you to sketch the line quickly and understand the function's rate of change. The slope highlights how steep the line is, and a positive or negative slope informs whether the line increases or decreases as \( x \) increases.

Function Verification

Verifying an inverse function ensures its accuracy and usefulness. To check if the inverse \( f^{-1}(x) \) is correct, you need to confirm that the original function and its inverse will bring you back to your starting value when applied successively.For \( f(x) = -3x + 9 \) and its inverse \( f^{-1}(x) = -\frac{1}{3}x + 3 \):

• Calculate \( f(f^{-1}(x)) \) and ensure it results in \( x \). When you substitute \( f^{-1}(x) \) into \( f(x) \), simplify to return to \( x \).
• Similarly, calculate \( f^{-1}(f(x)) \) and verify it leads to \( x \).

When both tests output \( x \), you confirm that \( f^{-1}(x) \) is the correct inverse of \( f(x) \). This process not only checks if you've done the calculations right but also reinforces your understanding of how original and inverse functions interact.