Problem 26 - College Algebra

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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$ f(x)=\frac{4}{x}+9 $$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=\frac{4}{x}+9\) is \(f^{-1}(x) = \frac{4}{x-9}\). The original and inverse functions satisfy the conditions \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), verifying that the function found is indeed the inverse.

Step 1: Find the inverse of the function

The given function is \(f(x) = \frac{4}{x}+9\). First, swap 'x' and 'y' to find the inverse function i.e., \(x = \frac{4}{y}+9\). Secondly, solve for 'y'. This involves rearranging the equation to isolate 'y'. Subtraction of '9' from each side of the equation gives \(x-9 = \frac{4}{y}\). Multiplying each side by 'y' gives \(y(x-9) = 4\). Finally, dividing by \(x-9\) gives the equation of the inverse function as \(f^{-1}(x) = \frac{4}{x-9}\).

Step 2: Verify \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into the original function. Hence \(f(f^{-1}(x)) = f(\frac{4}{x-9}) = \frac{4}{\frac{4}{x-9}}+9\). After simplification, we get \(x\). Hence, \(f(f^{-1}(x)) = x\).

Step 3: Verify \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) into the inverse function. So, \(f^{-1}(f(x)) = f^{-1}(\frac{4}{x}+9) = \frac{4}{\frac{4}{x}+9-9}\). After simplification, we get \(x\). Hence, \(f^{-1}(f(x)) = x\).

Key Concepts

One-to-One Functions

One-to-one functions are a fundamental concept in understanding inverses. In a one-to-one function, every output is associated with exactly one input. This means that for any two different inputs, the outputs will also be different.
To check if a function is one-to-one, consider doing a horizontal line test on its graph. If each horizontal line intersects the graph at most once, the function is one-to-one.
This property ensures that the function has an inverse, making it possible to "undo" the function. Since the given function, \(f(x) = \frac{4}{x} + 9\), is noted as one-to-one, we can proceed to find an inverse for it.
Remember, without being one-to-one, a function cannot have an inverse, because you would not be able to backtrack uniquely from an output to a single input.

Verification of Inverse

Verifying the inverse of a function is crucial to ensure that the operations carried out are correct. There are two main checks for verification:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

For a function and its inverse, these checks essentially confirm that starting with any input, running it through the function and then the inverse (or vice versa) will return you to the original input.
In our example, substitute the inverse \(f^{-1}(x) = \frac{4}{x-9}\) back into the original function \(f(x) = \frac{4}{x} + 9\). Simplify to show it equals \(x\) ensures the first verification is satisfied. Similarly, substituting the original function into the inverse and simplifying to \(x\) satisfies the second verification, resulting in a double confirmation that the inverse is accurate.

Function Inversion Steps

Inverting a function involves several logical steps which transform the original equation to its inverse. This process is systematic, allowing you to confidently derive inverse functions. Here are the basic steps:

• Swap 'x' and 'y': Start by changing the roles of \(x\) and \(y\) in the function equation. If \(f(x) = y\), then for the inverse, consider \(x = \frac{4}{y} + 9\).
• Solve for 'y': Rearrange the equation to isolate \(y\). This might involve operations like subtraction, multiplication, or division. As given \(x - 9 = \frac{4}{y}\), transitioning steps lead to \(y(x-9) = 4\) and finally \(y = \frac{4}{x-9}\).

Once you have \(y\) isolated, you've found your inverse function \(f^{-1}(x)\). Re-verifying, as shown in the next section, ensures this inverted function works correctly with the original.