https://file.aiquickdraw.com/mathbot/file/1725517319852ivhnf542.jpg

The image you've provided contains a series of mathematical functions and operations that need to be calculated. I'll go through the problems step by step.

### Provided Functions:
1. \( f(x) = x^2 + 1 \)
2. \( g(x) = 2x - 1 \)
3. \( h(x) = x + 1 \)

### Problems:
The problems listed involve various operations and compositions of the given functions. Let's solve them one by one.

### a. \( (f \circ g)(x) = f(g(x)) \)
- Substitute \( g(x) \) into \( f(x) \):
\[
g(x) = 2x - 1
\]
\[
f(g(x)) = f(2x - 1) = (2x - 1)^2 + 1
\]
- Expand the square:
\[
(2x - 1)^2 = 4x^2 - 4x + 1
\]
- Add 1 to the result:
\[
f(g(x)) = 4x^2 - 4x + 2
\]

### b. \( (g \circ f)(x) = g(f(x)) \)
- Substitute \( f(x) \) into \( g(x) \):
\[
f(x) = x^2 + 1
\]
\[
g(f(x)) = g(x^2 + 1) = 2(x^2 + 1) - 1
\]
- Simplify:
\[
g(f(x)) = 2x^2 + 2 - 1 = 2x^2 + 1
\]

### c. \( f(g(1)) \)
- Find \( g(1) \) first:
\[
g(1) = 2(1) - 1 = 1
\]
- Now, find \( f(1) \):
\[
f(1) = 1^2 + 1 = 2
\]
So, \( f(g(1)) = 2 \).

### d. \( g(f(1)) \)
- Find \( f(1) \) first:
\[
f(1) = 1^2 + 1 = 2
\]
- Now, find \( g(2) \):
\[
g(2) = 2(2) - 1 = 3
\]
So, \( g(f(1)) = 3 \).

### e. \( f(f(x)) \)
- Substitute \( f(x) \) into itself:
\[
f(f(x)) = f(x^2 + 1) = (x^2 + 1)^2 + 1
\]
- Expand:
\[
(x^2 + 1)^2 = x^4 + 2x^2 + 1
\]
- Add 1 to the result:
\[
f(f(x)) = x^4 + 2x^2 + 2
\]

### f. \( g(g(x)) \)
- Substitute \( g(x) \) into itself:
\[
g(g(x)) = g(2x - 1) = 2(2x - 1) - 1 = 4x - 2 - 1
\]
- Simplify:
\[
g(g(x)) = 4x - 3
\]

### g. \( (f \circ h)(x) = f(h(x)) \)
- Substitute \( h(x) \) into \( f(x) \):
\[
h(x) = x + 1
\]
\[
f(h(x)) = f(x + 1) = (x + 1)^2 + 1
\]
- Expand:
\[
(x + 1)^2 = x^2 + 2x + 1
\]
- Add 1 to the result:
\[
f(h(x)) = x^2 + 2x + 2
\]

### h. \( (g \circ f \circ h)(x) = g(f(h(x))) \)
- Substitute \( h(x) = x + 1 \) into \( f(x) \) first:
\[
f(h(x)) = (x + 1)^2 + 1 = x^2 + 2x + 2
\]
- Now, substitute into \( g(x) \):
\[
g(f(h(x))) = g(x^2 + 2x + 2) = 2(x^2 + 2x + 2) - 1
\]
- Simplify:
\[
g(f(h(x))) = 2x^2 + 4x + 4 - 1 = 2x^2 + 4x + 3
\]

These are the solutions to the given problems. Would you like further details on any of these steps or have any additional questions?

### Related Questions:
1. What is the difference between a composition of functions and a regular function operation?
2. How would the results change if \( f(x) = x^2 \) instead of \( x^2 + 1 \)?
3. How can you verify the results of a composition of functions using specific values of \( x \)?
4. What are the properties of function compositions?
5. How can compositions of functions be used in real-life applications?

### Tip:
When dealing with compositions of functions, always substitute the inner function into the outer one carefully, and simplify the expressions step by step to avoid errors.

Learn how to solve various function compositions, such as (f ∘ g)(x) and (g ∘ f)(x), using the provided quadratic and linear functions f(x) = x^2 + 1, g(x) = 2x - 1, and h(x) = x + 1. This problem includes step-by-step solutions for students in Grades 10-12.

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