Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=6 x$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 6x\) is \(f^{-1}(x) = x/6\). The properties \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\) both hold true.

Step 1: Find the Inverse Function

The function given is \(f(x)=6x\). For finding the inverse, one can replace \(f(x)\) with \(y\), then swap \(x\) and \(y\) and solve for \(y\). So, now we have \(y=6x\). Swap \(x\) and \(y\) to get \(x=6y\). Solving for \(y\) gives us \(y=x/6\). Therefore, \(f^{-1}(x)=x/6\).

Step 2: Verify the First Property

We have to verify the property \(f(f^{-1}(x))=x\). Substituting \(f^{-1}(x)\) into \(f(x)\), we have \(f(f^{-1}(x))=f(x/6)=6*(x/6)=x\). So the first property is verified.

Step 3: Verify the Second Property

Now, we need to verify the second property, \(f^{-1}(f(x))=x\). Substituting \(f(x)\) into \(f^{-1}(x)\), we have \(f^{-1}(f(x))=f^{-1}(6x)=(6x)/6=x\). Hence, the second property also holds true.

Key Concepts

Inverse Function Verification

Understanding the concept of inverse functions is crucial in the realm of mathematics. An inverse function essentially reverses the operation of the original function. To verify that you have correctly found an inverse function, there are two critical properties that must be checked:

Firstly, when you apply the original function to its inverse function, the result should be the input value of the inverse function, mathematically denoted as \(f(f^{-1}(x)) = x\). Secondly, if you apply the inverse function to the original function, you should again get the input value of the original function, expressed as \(f^{-1}(f(x)) = x\).

These properties ensure that the original function and its inverse are perfect flips of each other on the coordinate plane. When you graph both functions, they will reflect across the line \(y = x\), solidifying their inverse relationship. If both verification properties hold true, you can confidently say that you have correctly determined the inverse function.

Algebraic Manipulation

Algebraic manipulation involves rearranging, simplifying, or rewriting expressions and equations using algebraic rules. This skill is foundational for finding the inverse function, among other things in mathematics. To find an inverse, we often start by interchanging the roles of \(x\) and \(y\) in an equation.

For instance, given a function like \(f(x) = 6x\), we begin by writing it as \(y = 6x\). To find its inverse, we swap \(x\) and \(y\) yielding \(x = 6y\), and then solve for \(y\) by algebraic manipulation: \(y = \frac{x}{6}\). This becomes the formula for the inverse function, \(f^{-1}(x)\).

Command over algebraic manipulation is indispensable when working with functions and their inverses. It allows students to move variables and terms around confidently and correctly, which is essential in verifying whether two functions are indeed inverses of each other.

Function Composition

Function composition is a pivotal concept when dealing with functions and their inverses. It involves applying one function to the result of another function. In the context of inverse functions, composing a function with its inverse - either way round - should yield the identity function that maps any input to itself.

The notation \(f(g(x))\) denotes the composition of \(f\) and \(g\), where the output of \(g(x)\) becomes the input to \(f\). In our example, confirming that \(f^{-1}(x) = \frac{x}{6}\) is truly the inverse of \(f(x) = 6x\) requires us to compose them and look for the identity function: \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). Both compositions should simplify to \(x\), which verifies the inverse relationship.

By grasping function composition, students are empowered to tackle more complex functions and transform them, allowing for deeper understanding and exploration in mathematics and its applications.