Question

In Exercises 13–20, find the inverse of the function. Then graph the function and its inverse. $$ f(x)=\frac{2}{3} x-\frac{1}{3} $$

Short answer

The inverse of the function \(f(x) = \frac{2}{3}x - \frac{1}{3}\) is \(f^{-1}(x) = \frac{3}{2}x + \frac{1}{2}\). When graphed, these two functions are symmetric about the line \(y=x\).

Step-by-step solution

Identify the Given Function

The function given is \(f(x) = \frac{2}{3}x - \frac{1}{3}\).

Find the Inverse

To find the inverse of a function, replace \(f(x)\) with \(y\), then swap \(x\) and \(y\), and solve for \(y\). Thus, \(y = \frac{2}{3}x - \frac{1}{3}\) becomes \(x = \frac{2}{3}y - \frac{1}{3}\). Solving this equation for \(y\) yields, \(y = \frac{3}{2}x + \frac{1}{2}\). This is the inverse of the original function. Note: it's represented as \(f^{-1}(x) = \frac{3}{2}x + \frac{1}{2}\).

Graph the Functions

On a graph, plot both \(f(x) = \frac{2}{3}x - \frac{1}{3}\) and its inverse \(f^{-1}(x) = \frac{3}{2}x + \frac{1}{2}\). You will notice that the two graphs are symmetric around the line \(y=x\), which is a graphical representation of the fact that a function and its inverse swap the roles of the input and output.

Key concepts

Graphing Functions

Understanding how to graph functions is a fundamental skill in mathematics. It starts with recognizing the general shape of the function based on its equation. The given function in our exercise,
\( f(x) = \frac{2}{3}x - \frac{1}{3} \),
is a linear function which means that its graph is a straight line. To graph this function, first, identify two points by choosing values for \(x\) and calculating the corresponding \(y\) values. Then plot these points on a Cartesian plane and draw a line through them. It's crucial to ensure that the scale on both axes is consistent to accurately reflect the function's rate of change or slope.

• The slope of this function is \( \frac{2}{3} \), indicating how steeply the line rises or falls as one moves along the x-axis.
• The y-intercept is \( -\frac{1}{3} \), which is where the line crosses the y-axis.

Inverse Functions

Inverse functions are a way to 'undo' the effect of the original function. They essentially reverse the roles of inputs and outputs. To find an inverse, we follow the algebraic process demonstrated in the exercise: switch \(x\) and \(y\) and then solve for \(y\) to express the new function in terms of \(x\).

• First, write the function as \(y = \frac{2}{3}x - \frac{1}{3}\).
• Then switch the variables to get \(x = \frac{2}{3}y - \frac{1}{3}\).
• Solve for \(y\), which then gives the inverse function \(f^{-1}(x) = \frac{3}{2}x + \frac{1}{2}\).

Graphing the inverse alongside the original function allows us to visually verify if we have done this process correctly, as the graphs will be symmetric about the line \(y=x\).

Algebraic Manipulation

Algebraic manipulation is a tool that helps us rearrange equations to highlight different features or solve for specific variables. In the context of finding an inverse function, we rely heavily on algebraic manipulation.

• Starting with the equation \(x = \frac{2}{3}y - \frac{1}{3}\), we add \(\frac{1}{3}\) to both sides to isolate the term with \(y\).
• Multiply both sides of the equation by \(\frac{3}{2}\) to solve for \(y\).

The step-by-step algebraic rearrangement is critical for both finding the inverse function and understanding the relationship between the variables within an equation.

Function Symmetry

Function symmetry refers to the balanced and mirror-like property that can occur in the graphs of certain pairs of functions, such as a function and its inverse. When we graph a function and its inverse, as we did with \(f(x)\) and \(f^{-1}(x)\) from our exercise, we expect the graphs to reflect each other across the line \(y=x\).

• This line, \(y=x\), serves as a mirror; each point \((a, b)\) on the original function should correspond to the point \((b, a)\) on the inverse function.
• The symmetry becomes a visual confirmation that we've successfully found the inverse: if our graphs do not exhibit this symmetry, we may have made an error in the process.

By examining function symmetry, we enhance our understanding of the intrinsic relationship between a function and its inverse.