f ( X ) = x +3 
G ( x ) = 2x+ 7 
h(x) = x square find (fog) oh = fo (goh)

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.

Explore function composition with examples of f(g(x)) and g(h(x)). Learn how to solve (f ◦ g)(x), (g ◦ h)(x), and (f ◦ (g ◦ h))(x) step-by-step with detailed explanations and examples. Suitable for advanced high school students and anyone interested in mathematics.

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