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

Hey there, math enthusiasts! Today, we're diving into a super cool topic: verifying inverse functions. You know, those pairs of functions that kind of "undo" each other? We've got two functions here, $f(x) = 5x - 25$ and g(x) = rac{1}{5}x + 5. Our mission, should we choose to accept it (and we totally should!), is to figure out which expression can prove that $g(x)$ is indeed the inverse of $f(x)$. Let's break it down, shall we?

The Nitty-Gritty of Inverse Functions

So, what exactly makes two functions inverses of each other? It's all about composition, my friends! If $g(x)$ is the inverse of $f(x)$, then when you plug $g(x)$ into $f(x)$, you should get $x$. And, importantly, when you plug $f(x)$ into $g(x)$, you should also get $x$. It's a two-way street, a perfect symmetrical relationship. Mathematically, this means two things must be true:

1. $f(g(x)) = x$
2. $g(f(x)) = x$

If either of these conditions holds true, then boom! You've got yourselves inverse functions. If both hold true, then you're golden. Now, let's get our hands dirty with our specific functions.

Our trusty functions are $f(x) = 5x - 25$ and g(x) = rac{1}{5}x + 5. We need to test which of the given options helps us verify their inverse relationship. Let's look at the options:

A. rac{1}{5}ig(rac{1}{5} x+5ig)+5 B. rac{1}{5}(5 x-25)+5 C. rac{1}{ig(rac{1}{5} x+5ig)}

Wait a sec, the question asks for which expression could be used to verify. This implies we need to perform a function composition and see if it simplifies to $x$. The options provided look like potential results of these compositions. Let's try composing them ourselves and see which option matches.

Testing the Waters: Composition Time!

Let's tackle the first condition: $g(f(x))$. This means we take the entire function $f(x)$ and substitute it wherever we see $x$ in $g(x)$.

$g(f(x)) = g(5x - 25)$

Now, remember g(x) = rac{1}{5}x + 5. So, we replace the $x$ in $g(x)$ with $(5x - 25)$:

g(f(x)) = rac{1}{5}(5x - 25) + 5

Let's simplify this bad boy:

g(f(x)) = rac{1}{5} imes 5x - rac{1}{5} imes 25 + 5 $g(f(x)) = x - 5 + 5$ $g(f(x)) = x$

Boom! We got $x$! This is fantastic news. Now, let's compare this result to our options. The expression we just worked with, rac{1}{5}(5x - 25) + 5, is exactly Option B!

Double-Checking: The Other Way Around

Just to be absolutely sure, and because it's good practice, let's check the other composition: $f(g(x))$. This means we take $g(x)$ and plug it into $f(x)$.

f(g(x)) = fig(rac{1}{5}x + 5ig)

Remember $f(x) = 5x - 25$. So, we replace the $x$ in $f(x)$ with ig(rac{1}{5}x + 5ig):

f(g(x)) = 5ig(rac{1}{5}x + 5ig) - 25

Let's simplify this one too:

f(g(x)) = 5 imes rac{1}{5}x + 5 imes 5 - 25 $f(g(x)) = x + 25 - 25$ $f(g(x)) = x$

Double boom! We got $x$ again! This confirms that $g(x)$ is indeed the inverse of $f(x)$. The expression that we used to verify this second composition was 5ig(rac{1}{5}x + 5ig) - 25.

Now, let's look back at the options provided in the original question. The question asks which expression could be used to verify. We found that performing $g(f(x))$ yielded the expression rac{1}{5}(5x-25)+5, which is Option B. We also found that performing $f(g(x))$ yielded 5(rac{1}{5}x+5)-25.

Let's analyze the given options again:

A. rac{1}{5}ig(rac{1}{5} x+5ig)+5 - This looks like we're plugging $g(x)$ into $g(x)$, which isn't how we verify inverses. B. rac{1}{5}(5 x-25)+5 - This is exactly what we got when we calculated $g(f(x))$. This expression, when simplified, equals $x$. Thus, it verifies the inverse relationship. C. rac{1}{ig(rac{1}{5} x+5ig)} - This looks like taking the reciprocal of $g(x)$. This is not related to verifying inverse functions.

Therefore, the expression that could be used to verify $g(x)$ is the inverse of $f(x)$ is the one that results from one of the compositions and simplifies to $x$. We clearly saw that rac{1}{5}(5x-25)+5 is the result of $g(f(x))$ and it equals $x$.

Key Takeaway

When you're asked to verify if two functions, say $f$ and $g$, are inverses of each other, you need to check if both $f(g(x)) = x$ and $g(f(x)) = x$. The question specifically asks for an expression that could be used for this verification. Option B, rac{1}{5}(5x - 25) + 5, directly arises from one of these compositions ($g(f(x))$) and, when simplified, results in $x$, thus confirming the inverse relationship. It's all about performing that composition and seeing if you land back at $x$. Keep practicing these composition skills, guys, and you'll be an inverse function pro in no time!

So, to recap, the correct expression that helps verify $g(x)$ is the inverse of $f(x)$ is Option B. It's the direct result of plugging $f(x)$ into $g(x)$, and as we saw, it simplifies beautifully to $x$. This is the essence of what it means for functions to be inverses – they cancel each other out perfectly!

Keep those mathematical minds sharp, and stay tuned for more awesome math explorations right here on Plastik Magazine!

Final Answer Breakdown

• The Problem: Given $f(x) = 5x - 25$ and g(x) = rac{1}{5}x + 5, find the expression to verify $g(x)$ is the inverse of $f(x)$.
• Inverse Function Rule: For $g(x)$ to be the inverse of $f(x)$, we must have $f(g(x)) = x$ AND $g(f(x)) = x$.
• Calculating $g(f(x))$:
  • Start with g(x) = rac{1}{5}x + 5.
  • Substitute $f(x) = 5x - 25$ for $x$: g(f(x)) = rac{1}{5}(5x - 25) + 5.
  • Simplify: rac{1}{5}(5x) - rac{1}{5}(25) + 5 = x - 5 + 5 = x.
• Comparing with Options: The expression rac{1}{5}(5x - 25) + 5 matches Option B. Since this composition results in $x$, it serves as a verification.
• Calculating $f(g(x))$ (for completeness):
  • Start with $f(x) = 5x - 25$.
  • Substitute g(x) = rac{1}{5}x + 5 for $x$: f(g(x)) = 5ig(rac{1}{5}x + 5ig) - 25.
  • Simplify: 5(rac{1}{5}x) + 5(5) - 25 = x + 25 - 25 = x.
• Conclusion: Both compositions result in $x$, confirming they are inverses. Option B is the direct expression derived from one of these key compositions that demonstrates this verification. You guys nailed it!