Question

In Exercises 13–20, find the inverse of the function. Then graph the function and its inverse. $$ f(x)=-3 x $$

Short answer

The inverse of the function \(f(x) = -3x\) is \(f^{-1}(x) = -\frac{1}{3}x\). The graph of the original function is a straight line sloping downwards and passing through the origin, and the graph of the inverse function is a straight line with a less steep slope, also through the origin.

Step-by-step solution

Find the Inverse Function

To find the inverse of a function, we need to switch the x and y and then to solve for y. In this case, our original function is \(y=-3x\). So, switch x and y to get \(x = -3y\) then solve for y. This gives \(y = -\frac{1}{3}x\). So, the inverse function is \(f^{-1}(x) = -\frac{1}{3}x\)

Graph the Original Function and Its Inverse

To graph the function and its inverse, remember, algebraic functions are reflections of each other across the line \(y=x\). Begin with graphing \(y = -3x\), a downward slope crossing the origin (0,0). Then graph the inverse \(y = -\frac{1}{3}x\), which is also a line through the origin, but has a more gentle slope.

Key concepts

Function Graphing

Graphing a function and its inverse helps us visualize the relationship between them. For any function and its inverse, their graphs will be reflections of each other across the line \(y = x\). This concept illustrates how switching the roles of \(x\) and \(y\) results in the inverse.

• When graphing the original function \(y = -3x\), notice that it is a straight line passing through the origin with a negative slope.
• The inverse function \(y = -\frac{1}{3}x\) is also a straight line through the origin, but the slope is less steep.
• Both lines are symmetric with respect to the line \(y = x\), showing the reflection property of inverses.

Understanding this symmetry can greatly help in verifying that you have correctly found the inverse. It also provides a neat way to check your work visually.

Algebraic Functions

Algebraic functions involve operations like addition, subtraction, multiplication, division, and sometimes powers and roots. They form the backbone of many mathematical models and are frequently used in calculus.

• The given function \(f(x) = -3x\) is an example of a linear algebraic function because it is expressed as a multiplication by a constant.
• To find the inverse of such a function, we flip \(x\) and \(y\) and solve for the new \(y\). This involves the basic operation of division in this case.
• Algebraic functions like these are important because they can be manipulated easily, and their inverses often have straightforward expressions.

Recognizing and working with algebraic functions is crucial for solving real-world problems and advanced mathematics.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It is a very useful form because it gives us a clear picture of how a line behaves.

• For \(f(x) = -3x\), it is in the format \(y = mx + b\) with a slope \(m = -3\) and y-intercept \(b = 0\).
• The slope tells us how steep the line is, and since \(-3\) is negative, the line descends from left to right.
• The y-intercept is 0, meaning the line crosses the y-axis at the origin.
• Similarly, for the inverse \(y = -\frac{1}{3}x\), the slope \(m = -\frac{1}{3}\) is less steep and still descends, showing the function's weaker rate of change.

This form simplifies graphing the function and its inverse while making it easy to compare their slopes and intercepts, enhancing our understanding of their relationship.