Problem 63 - Elementary and Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. \(f(x)=-\frac{2}{3} x+3\)

Step-by-Step Solution

Answer

The inverse of \( f(x) = -\frac{2}{3}x + 3 \) is \( f^{-1}(x) = -\frac{3}{2}x + \frac{9}{2} \).

Step 1: Understanding the Inverse

The inverse of a function switches the roles of inputs and outputs. To find the inverse, solve the equation by swapping the 'x' and 'f(x)' values in the original function, and then solve for the new variable.

Step 2: Swap and Solve for x

Start with the original function: \( f(x) = -\frac{2}{3}x + 3 \). Replace \( f(x) \) with \( y \): \( y = -\frac{2}{3}x + 3 \). Swap \( x \) and \( y \): \( x = -\frac{2}{3}y + 3 \). Solve for \( y \): Subtract 3 from both sides to get \( x - 3 = -\frac{2}{3}y \). Multiply both sides by \( -\frac{3}{2} \) to isolate \( y \): \( y = -\frac{3}{2}(x - 3) \). Simplifying gives the inverse function: \( f^{-1}(x) = -\frac{3}{2}x + \frac{9}{2} \).

Step 3: Equation for Line of Symmetry

The function and its inverse are symmetrical about the line \( y = x \). This line of symmetry can help ensure that the function and its inverse are correctly graphed.

Step 4: Graphing\( f(x), f^{-1}(x)\) and Line of Symmetry

Graph the original function \( f(x) = -\frac{2}{3}x + 3 \) and its inverse \( f^{-1}(x) = -\frac{3}{2}x + \frac{9}{2} \). On the same coordinate plane, graph the line of symmetry \( y = x \). Ensure all plots are precise to illustrate symmetry.

Key Concepts

Graphing Functions

When graphing a function, you are visually representing mathematical relationships on a coordinate plane. For linear functions like \( f(x) = -\frac{2}{3}x + 3 \), start by identifying the y-intercept: the point where the graph crosses the y-axis. Here, it is the point (0, 3). Next, use the slope \( -\frac{2}{3} \) to determine the direction in which the line moves.

• The slope indicates that for every 3 units you move horizontally, the line moves 2 units downwards.
• Use a ruler to draw a straight line through these points extending across the plane.

Once the function is graphed, repeat these steps for the inverse function \( f^{-1}(x) = -\frac{3}{2}x + \frac{9}{2} \). Graphing both functions on the same set of axes reveals their relationship and visually demonstrates how each input has a corresponding output in the other function.

Line of Symmetry

A line of symmetry in the context of functions is an imaginary line that divides a graph into two congruent halves. For functions and their inverses, the line of symmetry is important because it highlights the point-to-point correspondences between the two functions.

• This line usually takes the form of \( y = x \) for standard function inverses.
• When graphed, every point on one function reflects directly across this line to a corresponding point on the other function.

Drawing the line of symmetry on your graph ensures you can easily check that inverses are correctly plotted. It emphasizes the visual balance and equivalence between a function and its inverse.

Inverse Relationship

The inverse relationship signifies a functional dependence where the roles of input and output are interchanged. Calculating an inverse essentially involves reflecting a function over the line \( y = x \) which acts as a mirror.

• To find the inverse of a function, swap \( x \) and \( y \) and solve for the new y.
• This results in a new equation representing the inverse.

For \( f(x) = -\frac{2}{3}x + 3 \), the inverse function is \( f^{-1}(x) = -\frac{3}{2}x + \frac{9}{2} \). The inverse undoes the action of the original function, illustrating a reversible process. Visually, plotting both functions with their symmetry line enhances understanding of how inputs and outputs swap, marking the beauty of mathematical symmetry and function inversion.