Problem 42 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-\frac{3}{4} x+2$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=-\frac{3}{4}x+2\) is \(f^{-1}(x)=-\frac{4}{3}(x - 2)\). When graphed, these two functions will be reflections of one another across the line \(y=x\).

Step 1: Find the Inverse

For the function \(f(x)=-\frac{3}{4}x+2\), replace the function notation \(f(x)\) with \(y\). This yields \(y = -\frac{3}{4}x + 2\). Now, to find the inverse, switch \(y\) and \(x\) to get \(x = -\frac{3}{4}y + 2\). Solve for \(y\) by subtracting 2 from both sides, then multiplying both sides by \(-\frac{4}{3}\). So the inverse of \(f(x)\) is \(f^{-1}(x)=-\frac{4}{3}(x - 2)\).

Step 2: Graph the Original Function

The original function \(f(x)=-\frac{3}{4}x+2\) is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope is -3/4, and the y-intercept is 2. Start by plotting the y-intercept point (0,2). The slope -3/4 means 'move 3 units down for every 4 units to the right'. Use this to find more points on the graph and plot the line.

Step 3: Graph the Inverse Function

The inverse function \(f^{-1}(x)=-\frac{4}{3}(x - 2)\) can be rewritten in the form \(y = -\frac{4}{3}x + \frac{8}{3}\). Hence the slope is -4/3 and the y-intercept is 8/3. Start by plotting the y-intercept point (0,8/3). The slope -4/3 means 'move 4 units down for every 3 units to the right'. Use this to plot more points on the graph and sketch the line.

Key Concepts

Graphing Linear Functions

Graphing linear functions is a fundamental topic in algebra. To graph a linear function, we need to find two main elements:

• The slope of the line, which indicates how steep the line is and its direction (positive slope means the line goes upwards, negative slope means it goes downwards). It is denoted by m in the equation.
• The y-intercept, which is the point where the line crosses the y-axis. This point is given by b in the equation, corresponding to the value of the function when x is zero.

To draw the graph, start by plotting the y-intercept on the Cartesian plane. Then, from this point, use the slope to determine another point. For example, if the slope is -3/4, from the y-intercept point, move 3 units down and 4 units to the right.
Draw a line through the points, extending it in both directions to complete the graph.
This technique can be applied to both the original function and its inverse, making it easier to visualize the relationship between the two.

Slope-Intercept Form

The slope-intercept form is a popular method for expressing linear equations, denoted as y = mx + b.
This form is advantageous because it clearly displays the slope and the y-intercept, making it straightforward to graph the function.
In the equation:

• \( m \) stands for the slope, showing how rapidly and in what direction the line changes. A negative slope, such as -3/4, results in a downward slant to the right.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis. For the function \( f(x) = -\frac{3}{4}x + 2 \), the y-intercept is 2.

The slope-intercept form is essential for solving problems related to linear functions, including finding inverses and graphing.
By transforming the function into this form, identifying the slope and y-intercept becomes more manageable, aiding in the graphing process.

Function Notation

Function notation is a standard way of representing functions, using symbols like \( f(x) \), which reads as "f of x."
This notation provides a clear method for denoting functions and is especially useful in mathematics for identifying outputs corresponding to specific inputs.
For example:

• The function \( f(x) = -\frac{3}{4}x + 2 \) designates that "f" is the name of the function, "x" is the input variable, and the equation describes how to obtain the function's output.
• To find the inverse of a function, you switch the roles of x and y (or f(x)) and solve for the new y.

Function notation is not just limited to simple expressions but extends to more complex functions as well.
It is an effective way to clearly communicate the relationship between variables and is fundamental in calculus and higher-level mathematics.