Problem 61 - 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. \(f(x)=4 x+3\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x - 3}{4} \).

Step 1: Write the Function as an Equation

We start with the function given in the problem: \( f(x) = 4x + 3 \). To find its inverse, we first write it as an equation with \( y \) representing \( f(x) \):\[ y = 4x + 3 \].

Step 2: Swap Variables

To find the inverse, we exchange \( x \) and \( y \) in the equation, resulting in \[ x = 4y + 3 \]. This reflects the concept of finding the inverse by 'swapping' the roles of the dependent and independent variables.

Step 3: Solve for y

Next, solve the equation for \( y \) to isolate it on one side, which yields the inverse function. Start by subtracting 3 from both sides: \[ x - 3 = 4y \], then divide both sides by 4 to solve for \( y \): \[ y = \frac{x - 3}{4} \]. Thus, the inverse function is \( f^{-1}(x) = \frac{x - 3}{4} \).

Step 4: Graph the Functions and Line of Symmetry

Graph both the original function \( f(x) = 4x + 3 \) and its inverse \( f^{-1}(x) = \frac{x - 3}{4} \). The line of symmetry for any function and its inverse is the line \( y = x \). On a coordinate grid, plot the line \( y = x \), then plot the graphs for \( y = 4x + 3 \) and \( y = \frac{x - 3}{4} \). The points on these graphs should be symmetric with respect to the line \( y = x \).

Key Concepts

Graphing Functions

Graphing functions helps us visualize mathematical equations and their relationships. For the function given in the exercise, we start with the original function, which is linear and can be written as:

• \( y = 4x + 3 \)

To graph, choose values for \( x \) and calculate the corresponding \( y \) values. This process generates coordinate points that you can plot:

• Choose some example \( x \): -1, 0, 1, 2
• Calculate \( y \): \( y = 4(-1) + 3 = -1 \), \( y = 4(0) + 3 = 3 \), \( y = 4(1) + 3 = 7 \), \( y = 4(2) + 3 = 11 \)
• Plot the points: (-1, -1), (0, 3), (1, 7), (2, 11)

Draw a straight line through these points to fully graph the function. For the inverse, use similar steps with the inverse function:

• The inverse function: \( y = \frac{x - 3}{4} \)
• Calculate few points as example, plot, and connect them

Graphing creates a visual story, allowing you to see and confirm the symmetry with its inverse.

Line of Symmetry

The line of symmetry for a function and its inverse is always the line \( y = x \). This line plays a critical role, as it mirrors the original function and its inverse function.Visualizing this line helps:

• Understand the inherent reflections between a function and its inverse
• Ensure the correctness of your inverse function

Draw the line \( y = x \) on the same coordinate plane as your graphs. Here's how you can do it:

• Plot easy points that satisfy \( y = x \), like: (0,0), (1,1), (2,2)
• Draw a diagonal line that passes through these points

This diagonal line should serve as a mirror line between the two graphs. Any point on the original function should have a corresponding point on the inverse function directly opposite from this line.

Function Inversion

The process of finding an inverse function reverses the roles of the input and output. It's basically asking: given an output, what input could have led to it?Here's a step-by-step explanation:

• Start with the function definition set as an equation (\( y = 4x + 3 \))
• Swap the variables \( x \) and \( y \) to reverse their roles (\( x = 4y + 3 \))
• Solve this new equation for \( y \) to isolate it
• By subtracting 3 from both sides, then dividing by 4, the inverse emerges: \( y = \frac{x - 3}{4} \)

Consider what happens in the context of real-world problems. If the original function represented a conversion of length in inches to length in centimeters, the inverse would represent the conversion back to inches from centimeters.Understanding this concept opens the door to countless applications where reversing a process or operation is necessary. Function inversion not only reassures us of a function's behavior but also lets us map the results back to possible starting points.