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.