Problem 62 - Elementary and 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 function is \( f^{-1}(x) = 3x - 1 \).

Step 1: Understand the function

The function given is \( f(x) = \frac{x}{3} + \frac{1}{3} \). This is a linear function with a slope of \( \frac{1}{3} \) and a y-intercept of \( \frac{1}{3} \).

Step 2: Swap and Replace

To find the inverse, swap \( x \) and \( y \) in the function. Start with \( y = \frac{x}{3} + \frac{1}{3} \), then swap to become \( x = \frac{y}{3} + \frac{1}{3} \).

Step 3: Solve for y

Solve the equation \( x = \frac{y}{3} + \frac{1}{3} \) for \( y \):1. Subtract \( \frac{1}{3} \) from both sides: \( x - \frac{1}{3} = \frac{y}{3} \).2. Multiply both sides by 3 to solve for \( y \): \( y = 3x - 1 \). This is the inverse function, \( f^{-1}(x) = 3x - 1 \).

Step 4: Graph the function and its inverse

Graph \( f(x) = \frac{x}{3} + \frac{1}{3} \) and its inverse \( f^{-1}(x) = 3x - 1 \) on the same coordinate system. The function \( f(x) \) is a line with a slope of \( \frac{1}{3} \), while \( f^{-1}(x) \) is a line with a slope of 3.

Step 5: Draw the line of symmetry

The line of symmetry for a function and its inverse is \( y = x \). Draw this line on the graph to show that the function and its inverse are mirror images across this line.

Key Concepts

Linear Functions

A linear function is the simplest form of a function that captures the essence of a line in algebra and calculus. Linear functions are defined by the equation of the form \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. These functions plot as straight lines on a coordinate graph, indicating a constant rate of change.

Understanding the slope is crucial as it tells you how steep the line is. In our example, \( f(x) = \frac{x}{3} + \frac{1}{3} \), the slope is \( \frac{1}{3} \), meaning for every unit increase in \( x \), \( f(x) \) increases by \( \frac{1}{3} \).

• The slope \( m = \frac{1}{3} \) shows the rate of change is gentle.
• The y-intercept \( b = \frac{1}{3} \) represents the point where the line crosses the y-axis.

Linear functions are foundational in understanding more complex functions and have applications in modelling real-world situations like speed, cost, and time.

Graphing Functions

Graphing is a visual way of representing functions and involves plotting points that satisfy the function equation on a coordinate system. In graphing linear functions, such as \( f(x) = \frac{x}{3} + \frac{1}{3} \), you'll always get a straight line.

Follow these steps to graph a linear function:

• Identify the y-intercept and plot it on the y-axis. For \( f(x) \), the y-intercept is \( \frac{1}{3} \).
• Use the slope to find another point. From \( \frac{1}{3} \), move up 1 unit for every 3 units you move to the right to use the slope of \( \frac{1}{3} \).
• Draw the line through these points.

Similarly, graph its inverse, \( f^{-1}(x) = 3x - 1 \). Repeat the steps, but this time, the slope is 3, so for every unit you move right, go up 3 units.

Graphing allows you to visually compare the original function and its inverse, providing insight into their relationship through their intersecting points and slopes.

Symmetry in Graphs

Symmetry in graphs is a useful tool to understand the relationship between a function and its inverse. For any function and its inverse, there is always a symmetrical relationship about the line \( y = x \).

To understand symmetry further:

• The line \( y = x \) acts as a mirror line. Any point on the function \( f(x) \) is mirrored on \( f^{-1}(x) \), and vice versa.
• If \( (a, b) \) is a point on \( f(x) \), then \( (b, a) \) will be on \( f^{-1}(x) \); they are reflections across this line.

In graphing the given function \( f(x) = \frac{x}{3} + \frac{1}{3} \) and its inverse \( f^{-1}(x) = 3x - 1 \), ensure to draw \( y = x \) to highlight their reflection symmetry.

This line of symmetry helps verify the accuracy of inverses and provides an intuitive visual understanding of their relationship, showing how they "undo" each other.