Problem 51 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=3 x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The given function is a linear function, denoted as \( f(x) = 3x \). Our task is to find its inverse, which essentially means finding a function \( f^{-1}(x) \) such that \( f(f^{-1}(x)) = x \).

Step 2: Replace and Swap

To find the inverse function, start by replacing \( f(x) \) with \( y \) to get \( y = 3x \). Then, swap \( x \) and \( y \) to get the equation \( x = 3y \).

Step 3: Solve for the Inverse

Now, solve the equation \( x = 3y \) for \( y \). Divide both sides by 3 to get \( y = \frac{x}{3} \). Thus, the inverse function is \( f^{-1}(x) = \frac{x}{3} \).

Step 4: Graph the Functions

To graph \( f(x) = 3x \) and its inverse \( f^{-1}(x) = \frac{x}{3} \), plot both functions on the same graph. They are both straight lines. The function \( f(x) = 3x \) has a slope of 3, whereas \( f^{-1}(x) = \frac{x}{3} \) has a slope of \( \frac{1}{3} \). These lines will be symmetric with respect to the line \( y = x \).

Step 5: Verify the Solution Graphically

Check the correctness of the inverses by ensuring that the graph of \( f(x) \) and \( f^{-1}(x) \) are reflective over the line \( y = x \). This confirms that the functions are indeed inverses of each other.

Key Concepts

Linear Functions

Linear functions are among the simplest types of functions you will encounter in mathematics, characterized by a straight line graph. They have the general form \( f(x) = ax + b \), where \( a \) and \( b \) are constants, and \( a \) is the slope of the line while \( b \) is the y-intercept. In the function provided, \( f(x) = 3x \), the slope \( a \) is 3 and there is no y-intercept, meaning \( b = 0 \). The slope \( a \) determines the steepness and direction of the line:

• A positive slope results in a line that rises as it moves from left to right.
• A negative slope results in a line that falls as it moves from left to right.
• A slope of zero indicates a horizontal line with no rise or fall.

Understanding these characteristics helps in sketching the graph of a linear function effectively. The function \( f(x) = 3x \) rises steeply with a slope of 3. It's useful to remember these properties when graphing linear functions and their inverses, as they share similar linearity but differ in slope and direction.

Graphing Functions

Graphing functions allows us to visually understand the behavior of mathematical equations. When graphing the linear function \( f(x) = 3x \), it's essential to identify two main points: the y-intercept and the slope.

• The y-intercept is the point where the line crosses the y-axis. For \( f(x) = 3x \), this is the origin, \((0, 0)\).
• The slope tells us that for every unit the line moves horizontally, it moves three units vertically. Therefore, another point would be \((1,3)\).

By plotting these points and connecting them with a straight line, you form the graph of \( f(x) = 3x \).Now, for the inverse function \( f^{-1}(x) = \frac{x}{3} \), plot key points such as \((0,0)\) and \((3,1)\), where the line moves one unit up for every three units to the right.The graphs of \( f(x) \) and \( f^{-1}(x) \) will intersect at the origin and are symmetric about the line \( y = x \). This symmetry visually confirms that these functions are indeed inverses.

Function Composition

Function composition involves combining two functions to form a new function. For two given functions \( f \) and \( g \), the composition \( f(g(x)) \) means you first apply \( g \) to \( x \), then \( f \) to the result. To explore inverse functions, understanding function composition is vital. An inverse function \( f^{-1} \) is uniquely defined such that:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

This property means that applying a function followed by its inverse returns the original value, effectively canceling each other out.In our exercise, if we compose \( f(x) = 3x \) and its inverse \( f^{-1}(x) = \frac{x}{3} \), we get:

• \( f(f^{-1}(x)) = f \left( \frac{x}{3} \right) = 3 \cdot \frac{x}{3} = x \)
• \( f^{-1}(f(x)) = f^{-1}(3x) = \frac{3x}{3} = x \)

These compositions confirm that \( f(x) \) and \( f^{-1}(x) \) properly undo each other, illustrating the essence of inverse functions through function composition.