Problem 62 - 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. $$ f(x)=\frac{x}{3}+\frac{1}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The provided function is a linear function, given by \( f(x) = \frac{x}{3} + \frac{1}{3} \). Our task is to find its inverse.

Step 2: Replace f(x) with y

To find the inverse, we first replace \( f(x) \) by \( y \). Thus, we have \( y = \frac{x}{3} + \frac{1}{3} \).

Step 3: Swap x and y

To find the inverse of the function, swap \( x \) and \( y \). So, the equation becomes \( x = \frac{y}{3} + \frac{1}{3} \).

Step 4: Solve for y

Solve the equation from Step 3 for \( y \). First, subtract \( \frac{1}{3} \) from both sides: \( x - \frac{1}{3} = \frac{y}{3} \). Then, multiply both sides by 3 to isolate \( y \): \( y = 3x - 1 \).

Step 5: Inverse Function

The inverse function is \( f^{-1}(x) = 3x - 1 \). Now we have the inverse of the original function.

Step 6: Graphing the Functions and Line of Symmetry

Plot both \( f(x) = \frac{x}{3} + \frac{1}{3} \) and \( f^{-1}(x) = 3x - 1 \) on the same coordinate system. The line \( y = x \) will act as the line of symmetry between these two graphs.

Key Concepts

Understanding Linear Functions

A linear function is a type of function where each input (or x-value) is multiplied by a constant, and then a constant is added.
This creates a straight line when graphed on a coordinate plane, which is why it is called "linear." For example, the function in your exercise is given by \( f(x) = \frac{x}{3} + \frac{1}{3} \). Here, the variable \( x \) is divided by 3, and then 1/3 is added to the result.
This simple structure helps make predictions about things that change at a constant rate, like speed or cost.

• The slope of the linear function is a key feature, represented by the coefficient of \( x \). In \( f(x) = \frac{x}{3} + \frac{1}{3} \), it is \( \frac{1}{3} \).
• The y-intercept, where the line crosses the y-axis, is another important part. It's given as \( \frac{1}{3} \) in the function \( f(x) = \frac{x}{3} + \frac{1}{3} \). This point is reached when \( x = 0 \).

Linear functions are a great starting point for understanding more complex mathematical ideas because they help set the foundation for those concepts.

Graphing Linear and Inverse Functions

Graphing functions is like creating a visual map of how values change. It's important for understanding how the original and inverse functions behave and relate to each other.
To graph a linear function like \( f(x) = \frac{x}{3} + \frac{1}{3} \), you take different values of \( x \), calculate corresponding \( y \) values, and plot them on a graph as points. Connecting these points forms a straight line.

• Plotting \( f(x) \): Choose simple values for \( x \), let's say \( 0, 3, -3 \), plug them into the function to get \( y \)-values, and plot the points.
• For \( f^{-1}(x) = 3x - 1 \), repeat the process with different \( x \)-values, such as \( 0, 1, -1 \), to plot its graph.

Using the inverse function \( f^{-1}(x) \), we see the reversal of \( x \) and \( y \)'s roles. Graphing these provides a visualization that enhances understanding and shows the connection between both functions.

Line of Symmetry and Its Importance

The line of symmetry is a crucial concept when discussing functions and their inverses. In the context of graphing both a function and its inverse, the line of symmetry helps show how the two are reflections of each other across this line.
In your graph, the line of symmetry is the line \( y = x \). It acts as a mirror, indicating that every point on \( f(x) \) will have a corresponding point on \( f^{-1}(x) \) such that if you fold along \( y = x \), one will match perfectly over the other.

• The line \( y = x \) is a simple diagonal line, slanting up from left to right, passing through points where \( x = y \).
• This concept illustrates that if a point \((a, b)\) is on \( f(x) \), then a point \((b, a)\) is on \( f^{-1}(x) \).

Understanding the concept of symmetry in functions, particularly when they are inverses, provides a deeper comprehension of how these functions relate and are useful in solving real-world problems.