Problem 15 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=2 x+3$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = 2x + 3\) is \(f^{-1}(x) = \frac{x - 3}{2}\). The verification step has shown that both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold true, confirming that the identified inverse is indeed correct.

Step 1: Find the Inverse Function

Start with the given function \(f(x) = 2x + 3\). To find the inverse, swap x and y: \(x = 2y + 3\). Solve this equation for y to get the inverse function: 1. Subtract 3 from both sides: \(x - 3 = 2y\).2. Divide each side by 2: \(y = \frac{x - 3}{2}\).This is the inverse function, written as \(f^{-1}(x) = \frac{x - 3}{2}\).

Step 2: Verify the First Inverse Property

Substitute \(f^{-1}(x)\) into the original function \(f(x)\):\(f(f^{-1}(x)) = 2f^{-1}(x) + 3 = 2\left(\frac{x - 3}{2}\right) + 3\).This simplifies to \(x\), which verifies the first property.

Step 3: Verify the Second Inverse Property

To verify the second property, substitute \(f(x)\) into the inverse function \(f^{-1}(x)\):\(f^{-1}(f(x)) = \frac{f(x) - 3}{2} = \frac{2x + 3 - 3}{2}\).This simplifies to \(x\), which verifies the second property. Hence, the function \(f^{-1}(x) = \frac{x - 3}{2}\) is indeed the inverse of the function \(f(x) = 2x + 3\).

Key Concepts

One-to-One Functions

When dealing with functions in precalculus, understanding the concept of one-to-one functions is crucial. A function is considered one-to-one if each input (or x-value) corresponds to exactly one unique output (or y-value), and vice versa.

In other words, no two different input values will produce the same output value. This uniqueness is important when it comes to inverse functions because only one-to-one functions have true inverses that are also functions. To check if a function is one-to-one, you can use the Horizontal Line Test. If any horizontal line crosses the graph of a function at most once, the function is one-to-one.

Finding Inverse Functions

To find an inverse function, sometimes denoted as f^-1(x), you'll typically start by writing the original function as an equation, like y = f(x). The subsequent steps involve algebraic manipulations:

• Swap 'x' and 'y' in the equation.
• Solve the swapped equation for 'y'.
• Replace 'y' with f^-1(x) to express the inverse function.

During the process, ensure that every step is inversible, meaning you can undo it. If any step is not reversible (such as squaring and square rooting, in which additional constraints must be considered), the function in question may not allow for a straightforward inverse.

Verifying Inverse Functions

Once you think you have found an inverse function, it's not enough to just claim it—the inverse needs to be verified. To do this, you'll employ function composition in two ways:

• By showing that f(f^-1(x)) = x for all x in the domain of f^-1.
• By proving f^-1(f(x)) = x for all x in the domain of f.

If both of these compositions result in 'x', your inverse function has been successfully verified. This ensures that the original function and its supposed inverse 'undo' each other, which is exactly the behavior we expect from inverse pairs.

Function Composition

Function composition is a valuable tool when working with functions, including verifying inverse functions. Composition involves applying one function to the results of another. In notation, composing f with g is written as f(g(x)), which is read as 'f of g of x'.

It's essential to perform composition correctly to verify that two functions are inverses of each other. Additionally, understanding composition can help you find composite functions' domains and ranges, recognize patterns, and simplify complex functions into more manageable parts. Remember that the order in which you compose functions matters: f(g(x)) may not equal g(f(x)).