To solve for [tex]\((g \circ f)(x)\)[/tex], which means evaluating [tex]\(g(f(x))\)[/tex], we will follow a step-by-step approach. Here are the steps:
1. Identify the Functions:
- [tex]\(f(x) = 8x - 8\)[/tex]
- [tex]\(g(x) = \sqrt{x + 4}\)[/tex]
2. Compose the Functions:
- To find [tex]\((g \circ f)(x)\)[/tex], we need to substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex].
- This means we will substitute [tex]\(8x - 8\)[/tex] into [tex]\(\sqrt{x + 4}\)[/tex].
3. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
- Since [tex]\(f(x) = 8x - 8\)[/tex], we replace [tex]\(x\)[/tex] in [tex]\(g(x)\)[/tex] with [tex]\(8x - 8\)[/tex].
- This gives us [tex]\(g(f(x)) = \sqrt{(8x - 8) + 4}\)[/tex].
4. Simplify Inside the Square Root:
- Combine like terms inside the square root: [tex]\(8x - 8 + 4\)[/tex].
- This simplifies to [tex]\(8x - 4\)[/tex].
5. Write the Final Form:
- Therefore, [tex]\((g \circ f)(x) = \sqrt{8x - 4}\)[/tex].
To provide a concrete example, we can evaluate this for a specific value, say, [tex]\(x = 2\)[/tex]:
1. Calculate [tex]\(f(2)\)[/tex]:
- [tex]\(f(2) = 8(2) - 8 = 16 - 8 = 8\)[/tex].
2. Calculate [tex]\(g(f(2))\)[/tex]:
- Substitute [tex]\(f(2) = 8\)[/tex] into [tex]\(g(x)\)[/tex].
- [tex]\(g(8) = \sqrt{8 + 4} = \sqrt{12}\)[/tex].
3. Simplify the Result:
- [tex]\(\sqrt{12}\)[/tex] can be simplified to [tex]\(2\sqrt{3}\)[/tex] if needed.
- We'll leave it in decimal form for accuracy, [tex]\(\sqrt{12} \approx 3.464\)[/tex].
So, [tex]\((g \circ f)(2) = \sqrt{12} \approx 3.464\)[/tex].
Therefore, [tex]\((g \circ f)(2) = \sqrt{12} \approx 3.464\)[/tex], which aligns with the previously provided numerical result.
