To find the composite function $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$. That is:

$(f \circ g)(x) = f(g(x))$

Given: $f(x) = \sqrt{x} + 7 \quad \text{and} \quad g(x) = 8x - 11$

Step 1: Substitute $g(x)$ into $f(x)$

Replace $x$ in $f(x)$ with $g(x)$: $f(g(x)) = \sqrt{g(x)} + 7$

Substitute $g(x) = 8x - 11$: $f(g(x)) = \sqrt{8x - 11} + 7$

Final Composite Function:

$(f \circ g)(x) = \sqrt{8x - 11} + 7$

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Would you like a step-by-step example with a specific $x$ value to further explain this? Let me know! 😊

Related Questions:

1. What is $(g \circ f)(x)$ for the same $f(x)$ and $g(x)$?
2. How do we find the domain of $(f \circ g)(x)$?
3. What happens if we compose these functions in the reverse order?
4. Can $(f \circ g)(x)$ ever equal zero? Solve for $x$.
5. How do you graph $(f \circ g)(x)$ effectively?

Tip:

Always check the domain of the composite function after finding it, as the domain of $g(x)$ and the square root in $f(x)$ can restrict the overall domain.