Problem 23 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ f(x)=\frac{5}{8} x $$

Step-by-Step Solution

Answer

The inverse of \( f(x) = \frac{5}{8}x \) is \( f^{-1}(x) = \frac{8}{5}x \).

Step 1: Understand the Function

The function given is a linear function, which means its graph will be a line. The function is defined as \( f(x) = \frac{5}{8} x \). To find the inverse, you need to switch the roles of \( x \) and \( y \) where \( y = f(x) \).

Step 2: Swap and Solve for New y Variable

Start by setting \( y = \frac{5}{8}x \) which represents the original function. To find the inverse, swap \( x \) and \( y \), giving us \( x = \frac{5}{8} y \). Solve for \( y \) by multiplying both sides by \( \frac{8}{5} \), yielding \( y = \frac{8}{5}x \). The inverse function is therefore \( f^{-1}(x) = \frac{8}{5}x \).

Step 3: Graph the Functions

Graph the original function \( f(x) = \frac{5}{8}x \) as a line with a slope of \( \frac{5}{8} \) through the origin. Next, graph the inverse function \( f^{-1}(x) = \frac{8}{5}x \), which is also a line through the origin but with a slope of \( \frac{8}{5} \). The line \( y=x \) should be the line of reflection, showing that the original and inverse functions are symmetric about this line.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions in mathematics. They are characterized by a straight line when graphed on a coordinate plane. A linear function can be written in the form:
\[ f(x) = ax + b \]
where \(a\) and \(b\) are constants. The constant \(a\) is the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.

• The slope, \(a\), tells us how steep the line is. A larger slope means a steeper line.
• The y-intercept, \(b\), provides the starting point of the line on the y-axis.

In the exercise, the linear function \( f(x)=\frac{5}{8}x \) has a slope of \(\frac{5}{8}\) and no y-intercept, indicating the line passes through the origin \((0,0)\). Understanding this form is essential when working with or graphing linear equations.

Graph of a Function

The graph of a function is a visual representation of all the possible outputs of a function for inputs in its domain. For linear functions, like \(f(x) = \frac{5}{8}x\), this graph is a straight line. To graph a linear function:

• Identify the slope and y-intercept. For our function, the slope is \(\frac{5}{8}\), and the y-intercept is 0.
• Begin at the y-intercept \((0,0)\).
• Use the slope to determine the rise over run. From the y-intercept, move up 5 units and over 8 units.

Plotting these points gives the line of the original function. The inverse function \(f^{-1}(x) = \frac{8}{5}x\) can be graphed similarly but starts from the same point and adjusts according to its slope. Observing the symmetry about the line \(y = x\) confirms the proper graphing of a function and its inverse.

Slope-Intercept Form

The slope-intercept form is a convenient way to express linear equations and makes graphing straightforward. The general formula is:
\[ y = mx + c \]
Here, \(m\) represents the slope of the line, and \(c\) represents the y-intercept. This formula allows for quick identification of both the slope and the initial position on the graph.

• A positive slope \(m\) means the line will rise to the right.
• A negative slope \(m\) indicates the line will fall to the right.
• If \(c = 0\), the line will pass through the origin.

In our function, \(f(x)=\frac{5}{8}x\), the slope is \(\frac{5}{8}\) and there is no additional term, meaning the line crosses through the origin. The inverse, \(f^{-1}(x) = \frac{8}{5}x\), also fits the slope-intercept form, with a new slope \(\frac{8}{5}\) and also passing through the origin.