Problem 25 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x}\)

Step-by-Step Solution

Answer

Functions \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x}\) are inverses.

Step 1: Understand the Inverse Function Property

Two functions \(f\) and \(g\) are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for every \(x\) in the domain of \(g\) and \(f\) respectively.

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

First, substitute \(g(x)\) into \(f(x)\), i.e., \(f(g(x)) = f\left(\frac{1}{x}\right)\). Since \(f(x) = \frac{1}{x}\), this becomes \(f\left(\frac{1}{x}\right) = \frac{1}{(\frac{1}{x})} = x\). This confirms that \(f(g(x)) = x\).

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

Now substitute \(f(x)\) into \(g(x)\), i.e., \(g(f(x)) = g\left(\frac{1}{x}\right)\). Since \(g(x) = \frac{1}{x}\), this becomes \(g\left(\frac{1}{x}\right) = \frac{1}{(\frac{1}{x})} = x\). This confirms that \(g(f(x)) = x\).

Step 4: Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\) are true for all \(x\), the functions \(f\) and \(g\) are indeed inverses of each other.

Key Concepts

Inverse Function Property

In mathematics, the Inverse Function Property is a fundamental concept that helps us identify when two functions are inverses of each other. Suppose we have two functions, \(f\) and \(g\). To prove they are inverses, we need to show that:

• \(f(g(x)) = x\) for every \(x\) in the domain of \(g\)
• \(g(f(x)) = x\) for every \(x\) in the domain of \(f\)

This means applying the first function and then the second should return the original value.
In the example given, both \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x}\) satisfy this property.
Whenever you hear someone talking about inverses in mathematics, they often refer to functions that "undo" each other in this precise way. This brings us a deeper understanding of the function's behavior within its domain. Everyone learning the basics of calculus will likely confront this property often.

Function Composition

Function composition is a powerful tool in mathematics. It allows us to create a new function by combining two existing functions.
The process is often likened to nested operations, where one function is applied to the result of another.
To visualize, if you have a function \(f\) and a function \(g\), you can form the composite function \(f\circ g\) by applying \(g\) first, then \(f\). The notation \(f(g(x))\) is used to represent this process.
Here are some important points:

• The order of function composition matters! \(f(g(x))\) is not the same as \(g(f(x))\).
• Function composition is only possible when the range of the inside function matches the domain of the outside function.

In the original exercise, we used function composition to verify that \(f\) and \(g\) are indeed inverse functions.
This shows how interrelated the concepts of function composition and inverse functions are.

Domain of Functions

Understanding the domain of functions is crucial, especially when dealing with inverse functions and function composition.
The domain of a function is the set of all possible inputs for the function. For the function \(f(x) = \frac{1}{x}\), the domain is all real numbers except zero because division by zero is undefined.
Why is zero excluded?

• Division by zero does not yield a real number and leads to undefined mathematical expressions.

The same domain applies to \(g(x) = \frac{1}{x}\) in our exercise, reinforcing that when discussing inverse functions, understanding the domain restrictions is key.
When checking inverse functions, always pay close attention to their domains to ensure the operations are valid for all potential input values. This can help avoid incorrect conclusions about whether two functions are truly inverses of each other.