Problem 200 - Calculus Volume 1

Question

Use composition to determine which pairs of functions are inverses. \(\quad f(x)=8 x+3, g(x)=\frac{x-3}{8}\)

Step-by-Step Solution

Answer

The functions \(f(x)=8x+3\) and \(g(x)=\frac{x-3}{8}\) are inverses.

Step 1: Understand the Problem

To determine if two functions are inverses, we need to check if the composition of the functions results in the identity function. Specifically, for functions \(f(x)\) and \(g(x)\), they are inverses if \(f(g(x)) = x\) and \(g(f(x)) = x\). In this problem, our functions are \(f(x) = 8x + 3\) and \(g(x) = \frac{x - 3}{8}\).

Step 2: Compute f(g(x))

First, we substitute \(g(x) = \frac{x - 3}{8}\) into \(f(x)\) to compute \(f(g(x))\): \[ f(g(x)) = f\left(\frac{x - 3}{8}\right) = 8\left(\frac{x - 3}{8}\right) + 3 = x - 3 + 3 = x.\] This simplifies to \(x\), which is the identity function.

Step 3: Compute g(f(x))

Next, we substitute \(f(x) = 8x + 3\) into \(g(x)\) to compute \(g(f(x))\): \[ g(f(x)) = g(8x + 3) = \frac{(8x + 3) - 3}{8} = \frac{8x}{8} = x.\] This also simplifies to \(x\), which is again the identity function.

Step 4: Conclusion

Both \(f(g(x)) = x\) and \(g(f(x)) = x\) have been shown to equal the identity function. Therefore, the pair of functions \(f(x)\) and \(g(x)\) are indeed inverses of each other.

Key Concepts

Function Composition

Function composition is the process of combining two functions to form a new function. This is done by taking the output of one function and using it as the input for another. For example, if you have two functions, \( f(x) \) and \( g(x) \), the composition of \( f \) and \( g \) is written as \( (f \circ g)(x) \). This means you first apply \( g(x) \), then apply \( f \) to the result.

In our example, we need to check \( f(g(x)) = x \) and \( g(f(x)) = x \).

• For \( f(g(x)) \), substitute \( g(x) = \frac{x - 3}{8} \) into \( f(x) = 8x + 3 \).
• For \( g(f(x)) \), use the output of \( f(x) = 8x + 3 \) as the input for \( g(x) = \frac{x - 3}{8} \).

By ensuring both compositions simplify to \( x \), we find that \( f(x) \) and \( g(x) \) are indeed inverse functions.

Identity Function

The identity function is a very special kind of function. It is represented simply as \( I(x) = x \). This means whatever input you provide, the output will be exactly the same. The significance of the identity function in mathematics, especially when working with inverse functions, is that after a function and its inverse are composed, the result should be the identity function.

For example, validating whether \( f(x) \) and \( g(x) \) are inverses involves checking that both \( f(g(x)) = x \) and \( g(f(x)) = x \). This means:

• If you apply \( g \) on some input \( x \), and then \( f \) on \( g(x) \), you should get back \( x \),
• Similarly, applying \( f \) and then \( g \) should return to the same starting point, which is \( x \).

The identity function is essentially the mathematical way of saying "nothing has changed," emphasizing the relationship between a function and its inverse.

Checking Inverses

Checking whether two functions are inverses of each other can feel magical when you get the hang of it! It is about confirming that these two functions "undo" each other's work. Here's a simple checklist to verify inverses:

• Calculate \( f(g(x)) \) and ensure it equals \( x \).
• Calculate \( g(f(x)) \) and check if it results in \( x \).

If both conditions hold true, then you can comfortably say that the functions are inverses of each other.

In our given problem, by calculating \( f(g(x)) \) resulting in \( x \) and \( g(f(x)) \) also simplifying to \( x \), both conditions are satisfied fully.

This means the operations you perform by first substituting into one function and then into the other precisely cancel out, demonstrating the inverse relationship effectively.