Problem 61 - 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: Understand the Problem

We need to find the inverse of the function \( f(x) = 4x + 3 \). This means we want to express \( x \) in terms of \( y \), where \( y = f(x) \).

Step 2: Substitute and Re-arrange

Replace \( f(x) \) with \( y \). Therefore, \( y = 4x + 3 \). Swap \( x \) and \( y \) to find the inverse. Now, solve for \( y \):\[ x = 4y + 3 \].

Step 3: Solve for y

Subtract 3 from both sides: \[ x - 3 = 4y \]. Divide both sides by 4 to solve for \( y \): \[ y = \frac{x - 3}{4} \].

Step 4: Write the Inverse Function

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

Step 5: Graph the Functions

Graph \( f(x) = 4x + 3 \) and \( f^{-1}(x) = \frac{x-3}{4} \) on the same coordinate system. The original function is a line with a slope of 4 and a y-intercept at 3. The inverse function is a line with a slope of \( \frac{1}{4} \) and a y-intercept at \( -\frac{3}{4} \).

Step 6: Draw the Line of Symmetry

The line of symmetry for a function and its inverse is the line \( y = x \). Plot this line on the graph to visually confirm the symmetry between \( f(x) \) and \( f^{-1}(x) \).

Key Concepts

Function Graphing

Graphing functions is a powerful method to visualize mathematical relationships. By plotting the function on a coordinate system, we can see how the output varies with change in input. This leads us to a better understanding of the underlying patterns and behaviors of the function.
For the function \( f(x) = 4x + 3 \), it plots as a straight line due to its linear nature. To graph it, we start by identifying two key features: the slope and the y-intercept.

• The slope of 4 tells us that for each unit increase in \( x \), the \( y \) value increases by 4 units.
• The y-intercept at 3 is where the graph crosses the y-axis, indicating that when \( x = 0 \), \( y = 3 \).
• Similarly, the inverse function \( f^{-1}(x) = \frac{x-3}{4} \) is another straight line.
• This inverse has a slope of \( \frac{1}{4} \) and a y-intercept at \( -\frac{3}{4} \).

By plotting both lines on the same graph, we reveal the relationship between the function and its inverse.

Line of Symmetry

The line of symmetry plays a crucial role when discussing functions and their inverses. The line of symmetry in this context is the line where every point is equidistant to corresponding points on two symmetrical shapes or lines.
For a function and its inverse, this line is projected as \( y = x \). When graphing, this will be a 45-degree line that bisects the first and third quadrants of the Cartesian plane.
Adding this line to the graph with the original function \( f(x) \) and its inverse \( f^{-1}(x) \) highlights their symmetrical nature. It shows how each point \( (a, b) \) on \( f(x) \) maps to the point \( (b, a) \) on \( f^{-1}(x) \).
This visual guide not only verifies the correctness of the inverse function but also enriches understanding about

• how function and inverses interact.
• the geometric interpretation of inversion.
• ensuring your graph is accurate.

Linear Functions

Linear functions are one of the most fundamental types of functions, forming the building blocks for more complex mathematical concepts.
A linear function is any function that can be graphed as a straight line. The standard form is \( f(x) = mx + b \), where:

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

For \( f(x) = 4x + 3 \), the slope is 4, showing a relatively steep ascent, and the y-intercept is 3.
The characteristic of linearity is preserved even for the inverse \( f^{-1}(x) = \frac{x-3}{4} \), illustrating that inverse functions of linear functions are also linear.
Comprehending these key features aids in not only solving for inverses but also in predicting behaviors, transformations, and symmetries in graphs.