To solve the problem, let's first determine the compositions $f \circ f$ and $g \circ g$ by substituting the functions into each other.

1. Finding $f \circ f$:

Given: $f(x) = x^2 + 7$

The composition $f \circ f$ means applying $f(x)$ to itself: $(f \circ f)(x) = f(f(x))$

Substitute $f(x)$ into itself: $f(f(x)) = f(x^2 + 7)$

Now, replace $x$ in $f(x) = x^2 + 7$ with $x^2 + 7$: $f(x^2 + 7) = (x^2 + 7)^2 + 7$

Expand $(x^2 + 7)^2$: $(x^2 + 7)^2 = x^4 + 14x^2 + 49$

So: $f(x^2 + 7) = x^4 + 14x^2 + 49 + 7$

Simplify the expression: $f(f(x)) = x^4 + 14x^2 + 56$

Thus: $(f \circ f)(x) = x^4 + 14x^2 + 56$

2. Finding $g \circ g$:

Given: $g(x) = \frac{2}{5}x \quad \text{where } x \neq 0$

The composition $g \circ g$ means applying $g(x)$ to itself: $(g \circ g)(x) = g(g(x))$

Substitute $g(x)$ into itself: $g(g(x)) = g\left(\frac{2}{5}x\right)$

Now, replace $x$ in $g(x) = \frac{2}{5}x$ with $\frac{2}{5}x$: $g\left(\frac{2}{5}x\right) = \frac{2}{5} \left(\frac{2}{5}x\right)$

Simplify the expression: $g(g(x)) = \frac{4}{25}x$

Thus: $(g \circ g)(x) = \frac{4}{25}x$

Final Answers:

• $(f \circ f)(x) = x^4 + 14x^2 + 56$
• $(g \circ g)(x) = \frac{4}{25}x$

Would you like more details on any of these steps, or have any other questions?

Here are 5 related questions:

1. How do you find the domain of a composed function?
2. What happens to the domain of $f(x)$ when it’s composed with itself?
3. Can $g(x)$ be composed with itself if $x = 0$?
4. How does composition differ from function multiplication?
5. What is the difference between $f \circ g$ and $g \circ f$?

Tip: When composing functions, always ensure that the output of the first function falls within the domain of the second function.