Unlock Function Composition: Find $(g ext{ O } F)(a)=|a|-2$

Hey math whizzes! Today, we're diving deep into the fascinating world of function composition, a super cool concept where we essentially plug one function into another. Our mission, should we choose to accept it, is to find the exact pair of functions, let's call them $f(a)$ and $g(a)$, that makes the composite function $(g ext{ o } f)(a)$ equal to $|a|-2$. This isn't just about crunching numbers, guys; it's about understanding how functions interact and build upon each other. We'll be exploring different combinations and really getting to grips with what makes a composite function tick. So, grab your calculators, put on your thinking caps, and let's unravel this mathematical mystery together! This problem requires a solid understanding of how composite functions are formed and how to work backward or forward to find the correct components. We need to systematically check each option provided to see which one yields the desired result. This involves substituting the inner function $f(a)$ into the outer function $g(a)$ and then simplifying the expression to see if it matches $|a|-2$. It's a process of elimination and verification, and paying close attention to the details of each function is key to success. Let's break down the process and make sure we don't miss any crucial steps. Remember, the notation $(g ext{ o } f)(a)$ means $g(f(a))$. We're given a target output, $|a|-2$, and we need to find the input functions $f(a)$ and $g(a)$ that produce it. This can be a bit like solving a puzzle, where you have the final picture and need to figure out the individual pieces. We'll go through each option methodically, performing the composition and comparing the result with our target expression. This will ensure we arrive at the correct answer with confidence. So, let's get started on this exciting mathematical journey!

Option A: $f(a)=a^2-4$ and $g(a)=\sqrt{a}$

Alright, let's kick things off by examining Option A, where we have $f(a) = a^2-4$ and $g(a) = \sqrt{a}$. To find $(g ext{ o } f)(a)$, we need to substitute $f(a)$ into $g(a)$. That means wherever we see 'a' in $g(a)$, we're going to plug in the entire expression for $f(a)$. So, $g(f(a)) = g(a^2-4)$. Since $g(a) = \sqrt{a}$, we replace 'a' with $(a^2-4)$, giving us $g(a^2-4) = \sqrt{a^2-4}$. Now, we need to compare this result, $\sqrt{a^2-4}$, with our target expression, $|a|-2$. Do they match? Not even close, guys! The expression $\sqrt{a^2-4}$ involves a square root and depends on whether $a^2-4$ is non-negative. The absolute value function, $|a|-2$, behaves very differently. For instance, if $a=3$, then $|a|-2 = |3|-2 = 3-2 = 1$. But for Option A, $(g ext{ o } f)(3) = \sqrt{3^2-4} = \sqrt{9-4} = \sqrt{5}$, which is definitely not 1. Also, notice that $\sqrt{a^2-4}$ is only defined for $|a| less 2$, whereas $|a|-2$ is defined for all real numbers 'a'. This mismatch in definition and value clearly tells us that Option A is not the correct answer. We're looking for a perfect match, and this one just doesn't cut it. Keep those thinking caps on; we've got more options to explore!

Option B: $f(a)=\frac{1}{2} a-1$ and $g(a)=2 a-2$

Moving on to Option B, we're presented with $f(a) = \frac{1}{2}a - 1$ and $g(a) = 2a - 2$. Let's perform the composition $(g ext{ o } f)(a)$ by substituting $f(a)$ into $g(a)$. So, we have $g(f(a)) = g(\frac{1}{2}a - 1)$. Now, substitute $(\frac{1}{2}a - 1)$ into the expression for $g(a)$, which is $2a - 2$. This gives us $g(\frac{1}{2}a - 1) = 2(\frac{1}{2}a - 1) - 2$. Time to simplify this expression, folks! Distribute the 2: $2 \times \frac{1}{2}a - 2 \times 1 - 2 = a - 2 - 2$. Combining the constants, we get $a - 4$. Now, let's compare our result, $a-4$, with the target expression, $|a|-2$. Do they match? Nope! They are quite different. For example, if $a=5$, then $|a|-2 = |5|-2 = 5-2 = 3$. But for Option B, $(g ext{ o } f)(5) = 5-4 = 1$. Again, not a match. If $a=-5$, then $|a|-2 = |-5|-2 = 5-2 = 3$. But for Option B, $(g ext{ o } f)(-5) = -5-4 = -9$. Definitely not a match. The presence of the absolute value in our target function $|a|-2$ is a key feature that $a-4$ simply doesn't replicate. This shows us that Option B is also incorrect. Don't get discouraged, though; we're getting closer to finding the right pair!

Option C: $f(a)=3-3 a$ and $g(a)=4 a-5$

Let's tackle Option C, where $f(a) = 3-3a$ and $g(a) = 4a - 5$. We need to compute $(g ext{ o } f)(a)$, which means we'll substitute $f(a)$ into $g(a)$. So, $g(f(a)) = g(3-3a)$. Now, plug $(3-3a)$ into the expression for $g(a)$, which is $4a-5$: $g(3-3a) = 4(3-3a) - 5$. Let's simplify this expression, shall we? First, distribute the 4: $4 \times 3 - 4 \times 3a - 5 = 12 - 12a - 5$. Combine the constant terms: $12 - 5 - 12a = 7 - 12a$. So, for Option C, $(g ext{ o } f)(a) = 7 - 12a$. Now, compare this result, $7-12a$, with our target expression, $|a|-2$. Are they the same? Absolutely not! The expression $7-12a$ is a linear function that changes sign and value depending on 'a', but it doesn't involve an absolute value in the way $|a|-2$ does. For instance, if $a=3$, $|a|-2 = |3|-2 = 1$. But for Option C, $(g ext{ o } f)(3) = 7 - 12(3) = 7 - 36 = -29$. Not a match. If $a=-3$, $|a|-2 = |-3|-2 = 3-2 = 1$. But for Option C, $(g ext{ o } f)(-3) = 7 - 12(-3) = 7 + 36 = 43$. Clearly, $7-12a$ does not behave like $|a|-2$. Therefore, Option C is also not the correct answer. We're zeroing in on the solution, and only one option remains!

Option D: $f(a)=5+a^2$ and $g(a)=\sqrt{a-5}-2$

Finally, we arrive at Option D, where $f(a) = 5+a^2$ and $g(a) = \sqrt{a-5}-2$. Let's perform the composition $(g ext{ o } f)(a)$ by substituting $f(a)$ into $g(a)$. We have $g(f(a)) = g(5+a^2)$. Now, substitute $(5+a^2)$ into the expression for $g(a)$, which is $\sqrt{a-5}-2$. This looks like: $g(5+a^2) = \sqrt{(5+a^2)-5}-2$. Let's simplify the expression under the square root: $(5+a^2)-5 = 5+a^2-5 = a^2$. So the expression becomes $\sqrt{a^2}-2$. Now, here's a crucial step, guys: what is $\sqrt{a^2}$? By definition, the square root of a square is the absolute value of the number. That is, $\sqrt{a^2} = |a|$. Therefore, our composite function simplifies to $|a|-2$. Ta-da! This result, $|a|-2$, exactly matches our target expression! This means Option D is the correct answer. We found the pair of functions that satisfies the given condition! It's awesome when everything clicks into place like that, right? This problem highlights the importance of carefully evaluating composite functions and remembering key mathematical definitions, like $\sqrt{x^2} = |x|$. It's these details that often make the difference between getting the right answer and missing it. So, remember this when you're tackling similar problems involving function composition and square roots. Keep practicing, and you'll master these concepts in no time!