Problem 94 - A Graphical Approach to College Algebra

Question

Decide whether the pair of functions \(f\) and \(g\) are inverses. Assume axes have equal scales. CAN'T COPY THE GRAPH $$f(x)=-\frac{3}{11} x, \quad g(x)=-\frac{11}{3} x$$

Step-by-Step Solution

Answer

Yes, the functions are inverses.

Step 1: Definition of Inverse Functions

Two functions, \( f(x) \) and \( g(x) \), are inverses of each other if and only if the composition of \( f \) and \( g \) in both possible orders equals the identity function. This means \( f(g(x)) = x \) and \( g(f(x)) = x \).

Step 2: Compute \( f(g(x)) \)

To find \( f(g(x)) \), substitute \( g(x) = -\frac{11}{3}x \) into \( f(x) = -\frac{3}{11}x \): \[f(g(x)) = f\left(-\frac{11}{3}x\right) = -\frac{3}{11}\left(-\frac{11}{3}x\right). \] Simplify this expression: \[-\frac{3}{11} \times -\frac{11}{3}x = 1 \times x = x.\]Therefore, \( f(g(x)) = x \).

Step 3: Compute \( g(f(x)) \)

To find \( g(f(x)) \), substitute \( f(x) = -\frac{3}{11}x \) into \( g(x) = -\frac{11}{3}x \): \[g(f(x)) = g\left(-\frac{3}{11}x\right) = -\frac{11}{3}\left(-\frac{3}{11}x\right). \] Simplify this expression:\[-\frac{11}{3} \times -\frac{3}{11}x = 1 \times x = x.\]Therefore, \( g(f(x)) = x \).

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) = -\frac{3}{11}x \) and \( g(x) = -\frac{11}{3}x \) are indeed inverses of each other.

Key Concepts

Function Composition

Function composition is like the mathematical version of a relay race. In it, the output of one function becomes the input for another. When we say we're composing functions, we mean that we're connecting them together to create a chain of processes. Specifically, if you have two functions, say \( f(x) \) and \( g(x) \), you can combine them into a single function through composition.

• The notation \( f(g(x)) \) means "first apply \( g \), then \( f \)."
• Similarly, \( g(f(x)) \) indicates the operation where \( f \) goes first, followed by \( g \).
• The result of this operation is a new function that operates on an input \( x \).

This idea is useful in verifying if two functions are inverses. For functions \( f(x) \) and \( g(x) \) to be inverses, composing them in both ways should bring you back to where you started — in other words, produce the identity function. With the functions given in the problem \( f(x) = -\frac{3}{11}x \) and \( g(x) = -\frac{11}{3}x \), we found that both their compositions equal \( x \). This result means they're inverses of each other.

Identity Function

The identity function is a special concept in mathematics. It is like the equivalent of hitting a reset button in a process. No matter the input, the identity function outputs the same value; it's how it keeps things unchanged. Anyway, let's write it mathematically: the identity function looks like \( I(x) = x \).

• Its main characteristic is that it always returns the input unchanged.
• For any function \( f \), if \( f(g(x)) = x \) and \( g(f(x)) = x \), it's said that \( f \) and \( g \) are inverses because their compositions revert the transformations back to \( x \).
• In our case, this means opening and closing the door without actually changing its state.

In the exercise given, we used this property to verify the functions \( f(x) = -\frac{3}{11}x \) and \( g(x) = -\frac{11}{3}x \). Since both function compositions resulted in \( x \), which is the identity function, we concluded the functions are indeed inverses.

Linear Functions

Linear functions are like the mathematical equivalent of a perfectly straight line. They are straightforward and predictable, much like a ruler's edge. The general form is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) the y-intercept.

• These functions create straight lines when graphed on a coordinate plane.
• The slope \( m \) determines how steep the line is - a positive slope climbs upward to the right, while a negative slope descends.
• In cases where \( b = 0 \), the line passes through the origin, and these are the types we deal with when considering inverse linear functions.

In the example of \( f(x) = -\frac{3}{11}x \) and \( g(x) = -\frac{11}{3}x \), notice their structure closely fits that of linear functions without a y-intercept term \( b \). Their slopes, multiply to \( 1 \) when their respective fractions are composed, returning us to the identity function \( I(x) = x \). This feature is what confirms their inverse relationship.