Problem 13 - Precalculus Essentials

Question

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=2 x$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=2x\) is \(f^{-1}(x)=\frac{x}{2}\). Both conditions \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\) are satisfied, verifying that \(f^{-1}(x)=\frac{x}{2}\) is indeed the correct inverse function of \(f(x)=2x\).

Step 1: Finding the Inverse of the Function

An inverse function of \(f(x)=2x\) can be found by rearranging the equation to solve for \(x\). First, replace \(f(x)\) with \(y\) to get \(y = 2x\). Then, interchange \(x\) and \(y\) to find the inverse, which gives \(x = 2y\). Finally, isolate \(y\) by dividing both sides by 2, which gives \(y = \frac{x}{2}\). Therefore, the inverse of the function \(f(x)\) is \(f^{-1}(x)=\frac{x}{2}\).

Step 2: Verifying f(f^-1(x))=x

Testing the inverse is performed by showing that \(f(f^{-1}(x))=x\). Substitute \(f^{-1}(x)\) into \(f(x)\), hence:\[f(f^{-1}(x))=2(f^{-1}(x))=2 \times \frac{x}{2}=x\]. Hence, \(f(f^{-1}(x))=x\) is verified.

Step 3: Verifying f^-1(f(x))=x

The next step is to prove \(f^{-1}(f(x))=x\). Substitute \(f(x)\) into \(f^{-1}(x)\), hence: \[f^{-1}(f(x))=\frac{f(x)}{2}=\frac{2x}{2}=x\]. Hence, \(f^{-1}(f(x))=x\) is verified.

Key Concepts

One-to-One Functions

A one-to-one function is a special kind of function where each element in the domain maps to a unique element in the range. This characteristic is important because it ensures that the function has an inverse. To determine if a function is one-to-one, we use the horizontal line test. If any horizontal line crosses the graph of the function more than once, the function is not one-to-one.

Key properties of one-to-one functions include:

• Each input corresponds to one output, and each output corresponds to one input.
• The function is either entirely increasing or decreasing.
• An inverse function exists only for one-to-one functions.

In the given exercise, the function is expressed as \(f(x) = 2x\). This function is linear and strictly increasing, meaning it passes the horizontal line test easily.

Proofs of Inverse Functions

To find the inverse function, we generally solve the equation for the input variable. This involves swapping the variables and then rearranging the equation to isolate the original variable. For example, given \(f(x) = 2x\), we replace \(f(x)\) with \(y\) to get \(y = 2x\). By swapping \(x\) and \(y\), we get \(x = 2y\). Finally, we solve for \(y\) to obtain \(y = \frac{x}{2}\). Thus, the inverse function is \(f^{-1}(x) = \frac{x}{2}\).

The proof that this inverse function is correct involves two steps:

• Showing \(f(f^{-1}(x)) = x\): Substitute \(f^{-1}(x)\) into \(f(x)\). So, \(f(f^{-1}(x)) = 2(\frac{x}{2}) = x\).
• Showing \(f^{-1}(f(x)) = x\): Substitute \(f(x)\) into \(f^{-1}(x)\). So, \(f^{-1}(f(x)) = \frac{f(x)}{2} = \frac{2x}{2} = x\).

Both steps confirm that \(f(x)\) and \(f^{-1}(x)\) are truly inverses of each other.

Function Verification

Verifying a function and its inverse involves proving that the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) both yield \(x\). This is essential in validating that the functions are inverses. These verifications need to be true for all elements in the domain of \(f(x)\) and the range of \(f^{-1}(x)\).

For the exercise:\

• The verification \(f(f^{-1}(x)) = x\) shows that starting from an output \(x\) of the inverse function and applying the original function gives us back \(x\).
• Conversely, \(f^{-1}(f(x)) = x\) indicates that starting from an output of the original function and applying the inverse function also returns to \(x\).

These tests confirm the precise relationship between \(f(x)\) and \(f^{-1}(x)\), ensuring their validity as inverse functions.