Function Composition And Inverse: True Or False?

Oct 17, 2025 by ADMIN 49 views

Hey guys! Let's dive into a fun math problem involving functions, their inverses, and compositions. We're given two functions, $f(x)$ and $g(x)$ , and our mission is to determine whether certain statements about their inverses and compositions are true or false. So, grab your thinking caps, and let's get started!

Understanding the Functions

First, let's clearly define our functions. We have $f(x) = 2x - 5$ , which is a simple linear function. Then we have $g(x) = rac{3x + 1}{x - 2}$ , which is a rational function with a restriction that $x$ cannot be equal to 2 (since that would make the denominator zero, and we can't divide by zero, right?). These functions are the building blocks for our investigation into inverses and compositions.

When we talk about functions, it's super important to understand what they do. Think of a function like a machine: you put something in (an input, often represented by x), and the machine spits something out (an output, which is the value of the function at x, often written as f(x) or g(x)). For $f(x)$ , we're taking our input, multiplying it by 2, and then subtracting 5. For $g(x)$ , it's a bit more complex – we multiply the input by 3, add 1, and then divide the whole thing by the input minus 2. This foundational understanding is crucial for tackling the rest of the problem.

Finding the Inverse of $g(x)$

The first statement we need to evaluate is whether the inverse of $g(x)$ is given by $g^{-1}(x) = rac{2x + 1}{x - 3}$ . Now, what exactly is an inverse function? Simply put, the inverse function "undoes" what the original function does. If you feed an input x into a function and get an output y, then feeding y into the inverse function should give you back x. It's like reversing a process.

To find the inverse of a function, we typically follow these steps:

Replace $g(x)$ with $y$ : So, we rewrite $g(x) = rac{3x + 1}{x - 2}$ as $y = rac{3x + 1}{x - 2}$ .
Swap $x$ and $y$ : This is the key step in finding the inverse. We get $x = rac{3y + 1}{y - 2}$ .
Solve for $y$ : Now we need to isolate $y$ on one side of the equation. Let's multiply both sides by $(y - 2)$ to get rid of the fraction: $x(y - 2) = 3y + 1$ . This expands to $xy - 2x = 3y + 1$ .
Rearrange the equation to group the terms containing $y$ on one side: We move the $3y$ term to the left and the $-2x$ term to the right: $xy - 3y = 2x + 1$ .
Factor out $y$ : We can factor out a $y$ from the left side: $y(x - 3) = 2x + 1$ .
Divide to isolate $y$ : Finally, we divide both sides by $(x - 3)$ to get $y = rac{2x + 1}{x - 3}$ .

So, after all that algebra, we've found the inverse function! We found that $g^{-1}(x) = rac{2x + 1}{x - 3}$ . This matches the statement given in the problem. Therefore, the first statement is TRUE!

Analyzing the Composite Function $(f ullet g)(x)$

The second statement claims that the composite function $(f ullet g)(x) = rac{6x - 12}{x - 2}$ . What's a composite function, you ask? Think of it as plugging one function into another. $(f ullet g)(x)$ means we first apply the function $g$ to $x$ , and then we take the result and plug it into the function $f$ . In other words, we're evaluating $f(g(x))$ . The little circle symbol " $ullet$ " is just a shorthand way of writing this.

Let's break it down step by step:

Find $g(x)$ : We already know that $g(x) = rac{3x + 1}{x - 2}$ .
Substitute $g(x)$ into $f(x)$ : This means we replace the x in $f(x)$ with the entire expression for $g(x)$ . So, we get: $f(g(x)) = fig( rac{3x + 1}{x - 2}ig) = 2ig( rac{3x + 1}{x - 2}ig) - 5$ .
Simplify: Now we need to simplify this expression. First, distribute the 2: $2ig( rac{3x + 1}{x - 2}ig) = rac{6x + 2}{x - 2}$ . So, our expression becomes: $f(g(x)) = rac{6x + 2}{x - 2} - 5$ .
Combine terms: To subtract the 5, we need to write it as a fraction with a common denominator of $(x - 2)$ . We can write 5 as $rac{5(x - 2)}{x - 2} = rac{5x - 10}{x - 2}$ . Now we have: $f(g(x)) = rac{6x + 2}{x - 2} - rac{5x - 10}{x - 2}$ .
Perform the subtraction: Now we can subtract the numerators: $f(g(x)) = rac{(6x + 2) - (5x - 10)}{x - 2} = rac{6x + 2 - 5x + 10}{x - 2}$ .
Simplify further: Combine like terms in the numerator: $f(g(x)) = rac{x + 12}{x - 2}$ .

So, after all the simplification, we find that $(f ullet g)(x) = rac{x + 12}{x - 2}$ .

Now, let's compare this to the statement in the problem, which says $(f ullet g)(x) = rac{6x - 12}{x - 2}$ . These expressions are different! Therefore, the second statement is FALSE!

Final Verdict

Alright, we've done the math, and we've analyzed both statements. Here's our final answer:

The inverse of the function $g(x)$ is $g^{-1}(x) = rac{2x + 1}{x - 3}$ . TRUE
The composite function $(f ullet g)(x) = rac{6x - 12}{x - 2}$ . FALSE

So, there you have it! We successfully determined the truthfulness of the statements about the inverse and composite functions. Remember, the key is to break down the problem into smaller, manageable steps, understand the definitions, and carefully perform the algebra. Keep practicing, and you'll become a function whiz in no time!