Problem 60 - 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. \(f(x)=-3 x\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is linear, represented as \(f(x) = -3x\). This function has a slope of -3 and a y-intercept of 0. To find its inverse, we need to follow proper steps for manipulating linear equations.

Step 2: Replace and Solve for \(x\)

Begin by replacing \(f(x)\) with \(y\): \(y = -3x\). To find the inverse, we exchange \(x\) and \(y\) and solve for \(y\). Thus, we have \(x = -3y\). Solve for \(y\) by isolating it: \(y = \frac{-x}{3}\). The inverse function is \(f^{-1}(x) = \frac{-x}{3}\).

Step 3: Graph the Original Function

Plot the function \(f(x) = -3x\). This is a straight line that passes through the origin (0,0) with a slope of -3, meaning it goes down 3 units for every unit it moves to the right.

Step 4: Graph the Inverse Function

Graph the inverse function \(f^{-1}(x) = \frac{-x}{3}\). This line also passes through the origin but with a slope of \(-\frac{1}{3}\), so it goes down 1 unit for every 3 units it moves to the right.

Step 5: Draw the Line of Symmetry

The line of symmetry for the function and its inverse is the line \(y = x\). It is a diagonal line through the origin with a slope of 1, dividing the plane into two symmetric halves. Plot this line to show symmetry.

Key Concepts

Linear Equations

A linear equation is an equation in which the highest power of the variable is one, meaning it forms a straight line when graphed. Such equations take the general form of \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept. Linear equations are fundamental in coordinate geometry and involve simple algebraic manipulation.

To determine the inverse of a linear function like \( f(x) = -3x \), you first need to understand that the original function has a slope \( m = -3 \) and passes through the origin, meaning \( c = 0 \). Finding the inverse involves swapping \( x \) and \( y \) in the equation and solving for \( y \). This results in the inverse function \( f^{-1}(x) = \frac{-x}{3} \).

This process allows us to find a function that, when composed with the original function, results in the identity function \( f(f^{-1}(x)) = x \). Linear equations are instrumental in understanding many mathematical and real-world phenomena.

Graphing Functions

Graphing functions is a key visual method to understand mathematical relationships. To graph a function like \( f(x) = -3x \), start by plotting points. Since it passes through the origin (0,0), this serves as a starting point. From there, apply the slope of -3, which means for every 1 unit you move to the right, the line moves 3 units down.

Similarly, the inverse function \( f^{-1}(x) = \frac{-x}{3} \) is plotted by starting at the origin and applying its slope of \( -\frac{1}{3} \). This means for every 3 units you move to the right, you move 1 unit down. Functions are best understood when visualized, as graphs provide an immediate overview of the function's behavior.

Having a grasp of how to graph allows you to visually verify the symmetry and check solutions effectively. This skill becomes especially powerful in calculus and real-world modeling.

Line of Symmetry

The line of symmetry is an essential concept in understanding inverses of functions. It represents a line over which you can "fold" the graph, and both sides will match. In the context of graphing, the line of symmetry for a function and its inverse is often the line \( y = x \).

This line has a slope of 1 and passes through the origin, cutting the first and third quadrants symmetrically. When graphing a function and its inverse, draw the line \( y = x \) to see how each point on the original function maps to the corresponding point on the inverse function across this line.

The presence of a line of symmetry is a hallmark of inverses. Recognizing this feature can make it easier to understand and verify the transformation between functions and their inverses, improving understanding and accuracy in graphing.