To solve the composition of functions problem, we need to find:

1. $(f \circ g)(x)$ or $f(g(x))$
2. $(g \circ h)(x)$ or $g(h(x))$
3. $(f \circ (g \circ h))(x)$ or $f(g(h(x)))$

Step 1: Find $(f \circ g)(x)$

This means we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = x + 3$
• $g(x) = 2x + 7$

$(f \circ g)(x) = f(g(x)) = f(2x + 7)$

Now, substitute $g(x) = 2x + 7$ into $f(x)$:

$f(g(x)) = (2x + 7) + 3 = 2x + 10$

So, $(f \circ g)(x) = 2x + 10$.

Step 2: Find $(g \circ h)(x)$

This means we substitute $h(x)$ into $g(x)$.

Given:

• $h(x) = x^2$

$(g \circ h)(x) = g(h(x)) = g(x^2)$

Now, substitute $h(x) = x^2$ into $g(x)$:

$g(h(x)) = 2(x^2) + 7 = 2x^2 + 7$

So, $(g \circ h)(x) = 2x^2 + 7$.

Step 3: Find $(f \circ (g \circ h))(x)$

This means we substitute $(g \circ h)(x)$ into $f(x)$.

Using the result from Step 2:

$(f \circ (g \circ h))(x) = f(g(h(x))) = f(2x^2 + 7)$

Now, substitute $2x^2 + 7$ into $f(x)$:

$f(g(h(x))) = (2x^2 + 7) + 3 = 2x^2 + 10$

So, $(f \circ (g \circ h))(x) = 2x^2 + 10$.

Final Answers:

1. $(f \circ g)(x) = 2x + 10$
2. $(g \circ h)(x) = 2x^2 + 7$
3. $(f \circ (g \circ h))(x) = 2x^2 + 10$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the domain of the composite function $(f \circ g)(x)$?
2. How would the composition change if $h(x) = x^3$ instead of $x^2$?
3. Can you find the inverse of the function $f(x) = x + 3$?
4. What happens if you compose the functions in a different order, such as $(g \circ f)(x)$?
5. How do you determine if a function is one-to-one or onto?

Tip: When dealing with function compositions, always perform the substitution step carefully to avoid errors in simplification.