Problem 24 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ f(x)=\frac{1}{3} x+4 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 3x - 12 \). Graph both functions for verification.

Step 1: Understand Function Notation

The function given is \( f(x) = \frac{1}{3}x + 4 \), which is a linear function. We need to find its inverse and graph it along with the original function.

Step 2: Replace f(x) with y

Start by writing \( f(x) = \frac{1}{3}x + 4 \) as \( y = \frac{1}{3}x + 4 \), which will help in setting up the equation to find the inverse.

Step 3: Swap x and y

To find the inverse function, interchange \( x \) and \( y \) in the equation: \( x = \frac{1}{3}y + 4 \).

Step 4: Solve for y

Solve the equation \( x = \frac{1}{3}y + 4 \) for \( y \). First, subtract 4 from both sides to get \( x - 4 = \frac{1}{3}y \). Then multiply both sides by 3 to solve for \( y \): \( y = 3(x - 4) \).

Step 5: Express the Inverse Function

We now write the solution as the inverse function: \( f^{-1}(x) = 3x - 12 \). This function represents the inverse of the original function.

Step 6: Graph the Original and Inverse Functions

To graph \( f(x) = \frac{1}{3}x + 4 \), start with the y-intercept at \( y = 4 \) and a slope of \( \frac{1}{3} \). For the inverse \( f^{-1}(x) = 3x - 12 \), begin with the y-intercept at \( y = -12 \) and a slope of 3. Note that the lines will be reflections of each other over the line \( y = x \).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions and are characterized by their constant rate of change or slope. The general form of a linear function is expressed as: \( f(x) = mx + b \) where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept where the line crosses the y-axis.

In the exercise, we had the linear function \( f(x) = \frac{1}{3} x + 4 \). Here, the slope \( m \) is \( \frac{1}{3} \), indicating that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \). The y-intercept \( b \) is 4, meaning the line crosses the y-axis at 4.
Linear functions are always straight lines when graphed. This makes them easy to visualize and analyze. Understanding and working with these functions is a fundamental skill in algebra and calculus. Every inverse of a linear function is also a linear function, with the steepness of the slope effectively 'flipping'.

Graphing Functions

Graphing functions allows us to visualize their behavior and properties. In the exercise, we graph the original function and its inverse on the same set of axes.
To graph a linear function like \( f(x) = \frac{1}{3}x + 4 \):

• Start by plotting the y-intercept. For this function, it's (0, 4).
• Use the slope \( \frac{1}{3} \) to find another point. From (0, 4), move one unit to the right and up \( \frac{1}{3} \).
• Draw a straight line through these points.

The inverse function \( f^{-1}(x) = 3x - 12 \) is graphed similarly:

• Begin at the y-intercept, which is (0, -12).
• Use the slope of 3 to find another point. Move one unit to the right and three units up from (0, -12).
• Draw a straight line through these points.

The original function and its inverse are reflections across the line \( y = x \). This reflection property helps verify that the two functions are indeed inverses of each other.

Function Notation

Function notation is a concise way to express the operations applied to the input value \( x \) in mathematical functions. The notation \( f(x) \) signifies that \( f \) is a function of \( x \). It's a way to name and refer to functions that rely on input values to produce outputs.
Functions in this notation format clearly express input-output relationships. When we write \( f(x) = \frac{1}{3} x + 4 \), we identify that any specific \( x \) substituted into \( f \) will yield a corresponding y-value.
If we reverse the process to find an inverse function, like in our exercise, we switch the role of \( x \) and \( y \). The original function \( f \) can be flipped, and a new function \( f^{-1} \) is expressed.

• This means if \( f(a) = b \), the inverse function satisfies \( f^{-1}(b) = a \).

Function notation makes communication more efficient and is crucial for learning to manipulate functions, particularly in algebra and calculus. It allows mathematicians to easily discuss the transformations and behaviors of functions.