Verify Inverse Functions: F(x) And G(x) Explained

Hey guys! Ever stumbled upon a math problem that asks you to verify if one function is the inverse of another? It's a pretty common scenario, especially when dealing with functions like $f(x)=5x-25$ and g(x)=rac{1}{5}x+5. The big question here is: which expression can we use to prove that $g(x)$ is indeed the inverse of $f(x)$? Let's dive deep into this and break it down so it makes perfect sense.

Understanding Inverse Functions

First off, what exactly is an inverse function? Think of it like this: if a function takes you from point A to point B, its inverse function takes you back from point B to point A. They essentially undo each other's operations. Mathematically, if $g(x)$ is the inverse of $f(x)$, then two conditions must be met: $f(g(x)) = x$ and $g(f(x)) = x$. This is the golden rule, the ultimate test! It means that when you compose the functions in either order, you should end up with just '$x$' as your result. It’s like putting a key in a lock (the function) and then using the same key to unlock it (the inverse function) – you get back to where you started.

This concept is super important in various areas of mathematics, from algebra to calculus and beyond. Understanding how to find and verify inverse functions is a fundamental skill that opens doors to solving more complex problems. When we're given two functions, say $f(x)$ and $g(x)$, and we want to see if they're inverses, we need to plug one into the other and see if we get that magical '$x$' back. It’s a bit like a mathematical puzzle, and hitting that '$x$' is like solving it!

For our specific problem, we have $f(x) = 5x - 25$ and g(x) = rac{1}{5}x + 5. We need to find the expression that verifies $g(x)$ is the inverse of $f(x)$. This means we need to check if either $f(g(x)) = x$ or $g(f(x)) = x$. The options provided are A, B, C, and D. Let's analyze what each of these represents. Option A is rac{1}{5}ig(rac{1}{5}x+5ig)+5, which looks like we're plugging $g(x)$ into itself, not $f(x)$. Option C, rac{1}{ig(rac{1}{5}x+5ig)}, seems to be a reciprocal, not a composition. This leaves us with options that involve composing $f$ and $g$. The core idea is to substitute the entire $g(x)$ expression wherever you see '$x$' in the $f(x)$ expression, or vice versa. That's the process that will lead us to the verification.

The Verification Process: Step-by-Step

Alright, let's get down to business and perform the actual verification. We need to check the composition of $f(x)$ and $g(x)$. There are two ways to do this, and if either one results in '$x$', we've found our proof. Let's try the first way: calculating $f(g(x))$. This means we take the entire expression for $g(x)$, which is rac{1}{5}x + 5, and substitute it into $f(x)$ wherever we see '$x$'. Remember, $f(x) = 5x - 25$. So, f(g(x)) = fig(rac{1}{5}x + 5ig).

Now, we substitute: fig(rac{1}{5}x + 5ig) = 5ig(rac{1}{5}x + 5ig) - 25. Let's simplify this expression. First, distribute the 5: 5 imes rac{1}{5}x = x, and $5 imes 5 = 25$. So, the expression becomes $x + 25 - 25$. And what does $x + 25 - 25$ simplify to? It simplifies beautifully to just '$x$'. Bingo! We found it.

So, $f(g(x)) = x$. This result alone is sufficient to verify that $g(x)$ is the inverse of $f(x)$. We didn't even need to check the other composition, $g(f(x))$, though it's good practice to know how to do that too! The expression that represents this successful verification is 5ig(rac{1}{5}x + 5ig) - 25. Let's look at the options given:

A. rac{1}{5}ig(rac{1}{5}x+5ig)+5 - This is $g(g(x))$. Not what we want. B. rac{1}{5}(5x-25)+5 - This looks like $g(f(x))$. Let's check this one too, just for kicks! C. rac{1}{ig(rac{1}{5}x+5ig)} - This is the reciprocal of $g(x)$. Definitely not an inverse verification. D. This option usually represents a choice, like "Both A and B" or similar, or is just a placeholder. In this context, it's likely a distractor or represents an incorrect composition.

Let's evaluate option B, which is $g(f(x))$. This means we substitute $f(x)$ into $g(x)$. So, $g(f(x)) = g(5x - 25)$. Since g(x) = rac{1}{5}x + 5, we get: g(5x - 25) = rac{1}{5}(5x - 25) + 5. Distributing the rac{1}{5}: rac{1}{5} imes 5x = x, and rac{1}{5} imes -25 = -5. So, the expression becomes $x - 5 + 5$. And $x - 5 + 5$ simplifies to '$x$'.

So, both $f(g(x)) = x$ and $g(f(x)) = x$. This confirms that $g(x)$ is indeed the inverse of $f(x)$. The question asks which expression could be used to verify this. Both $f(g(x))$ and $g(f(x))$ are valid expressions for verification. Now we need to match these to the options.

Our calculation for $f(g(x))$ yielded 5ig(rac{1}{5}x + 5ig) - 25. This expression is not directly listed as an option, but it's the process of calculating it that matters. Let's re-examine the options in light of our findings.

Option B is rac{1}{5}(5x-25)+5. As we showed, this is the expression for $g(f(x))$, and it simplifies to '$x$'. Therefore, this expression could be used to verify that $g(x)$ is the inverse of $f(x)$.

What about option A? rac{1}{5}ig(rac{1}{5}x+5ig)+5. This is $g(g(x))$. If we simplify it: rac{1}{5}(rac{1}{5}x+5)+5 = rac{1}{25}x + 1 + 5 = rac{1}{25}x + 6. This is not '$x$', so A is incorrect.

Option C is rac{1}{ig(rac{1}{5}x+5ig)}. This is the reciprocal of $g(x)$, not a composition that verifies inverse functions. So, C is incorrect.

Since option B, rac{1}{5}(5x-25)+5, represents $g(f(x))$ which simplifies to '$x$', it is a valid expression used to verify that $g(x)$ is the inverse of $f(x)$.

Why Option B is the Correct Choice

The core principle for verifying inverse functions is the composition property: $f(g(x)) = x$ and $g(f(x)) = x$. We are looking for an expression that, when evaluated, proves this property. We found that $g(f(x))$ results in the expression rac{1}{5}(5x-25)+5. When this expression is simplified, it equals '$x$', thus verifying the inverse relationship.

Let's recap our verification steps to be absolutely crystal clear, guys. We were given $f(x)=5x-25$ and g(x)=rac{1}{5}x+5. We need to find an expression that proves $g(x)$ is the inverse of $f(x)$. This means we need to see if composing them gives us '$x$'.

We tested $f(g(x))$: f(rac{1}{5}x+5) = 5(rac{1}{5}x+5) - 25 = x + 25 - 25 = x. The expression that represents this composition is 5(rac{1}{5}x+5) - 25. This isn't option B.

Then we tested $g(f(x))$: g(5x-25) = rac{1}{5}(5x-25) + 5. This is exactly option B!

Let's simplify option B to confirm it equals '$x$': rac{1}{5}(5x-25)+5 = (rac{1}{5} imes 5x) - (rac{1}{5} imes 25) + 5 $= x - 5 + 5$ $= x$

Since the expression in option B, when simplified, equals '$x$', it serves as a direct verification that $g(x)$ is the inverse of $f(x)$. It demonstrates that applying $f$ first and then $g$ brings us back to the original input '$x$'. This is the fundamental definition of inverse functions in action, and option B perfectly captures one of the ways to show this.

It's crucial to understand that the question is asking for the expression that can be used for verification, not just the final result '$x$'. Option B is the unsimplified form of the composition $g(f(x))$, and evaluating it confirms the inverse relationship. Therefore, option B is the correct answer because it represents the composition $g(f(x))$, which simplifies to '$x$', thereby verifying that $g(x)$ is the inverse of $f(x)$. This method is fundamental to checking inverse relationships in algebra and is a key concept for anyone studying functions. Keep practicing these compositions, and you'll master inverse functions in no time!