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Let's solve the given problem step-by-step. The function is defined as \( f(x) = x^2 + 1 \). Now, we'll compute the values required in each part.

### (a) \( f(t+1) \)
We substitute \( t+1 \) into the function:
\[
f(t+1) = (t+1)^2 + 1
\]
Expand the square:
\[
f(t+1) = t^2 + 2t + 1 + 1 = t^2 + 2t + 2
\]

### (b) \( f(t^2 + 1) \)
Substitute \( t^2 + 1 \) into the function:
\[
f(t^2 + 1) = (t^2 + 1)^2 + 1
\]
First, expand \( (t^2 + 1)^2 \):
\[
(t^2 + 1)^2 = t^4 + 2t^2 + 1
\]
Now, add 1:
\[
f(t^2 + 1) = t^4 + 2t^2 + 1 + 1 = t^4 + 2t^2 + 2
\]

### (c) \( f(2) \)
Substitute 2 into the function:
\[
f(2) = 2^2 + 1 = 4 + 1 = 5
\]

### (d) \( 2f(t) \)
First, find \( f(t) \):
\[
f(t) = t^2 + 1
\]
Now, multiply by 2:
\[
2f(t) = 2(t^2 + 1) = 2t^2 + 2
\]

### (e) \( (f(t))^2 + 1 \)
Find \( (f(t))^2 \) first:
\[
f(t) = t^2 + 1
\]
Square it:
\[
(f(t))^2 = (t^2 + 1)^2 = t^4 + 2t^2 + 1
\]
Now, add 1:
\[
(f(t))^2 + 1 = t^4 + 2t^2 + 1 + 1 = t^4 + 2t^2 + 2
\]

### Summary of Results:
- (a) \( f(t+1) = t^2 + 2t + 2 \)
- (b) \( f(t^2 + 1) = t^4 + 2t^2 + 2 \)
- (c) \( f(2) = 5 \)
- (d) \( 2f(t) = 2t^2 + 2 \)
- (e) \( (f(t))^2 + 1 = t^4 + 2t^2 + 2 \)

Would you like more details or explanations on any part?

### Related Questions:
1. What happens if we differentiate \( f(x) = x^2 + 1 \)?
2. How does the function \( f(x) = x^2 + 1 \) change when shifted horizontally by 1?
3. What is the domain and range of the function \( f(x) = x^2 + 1 \)?
4. How do you compute \( f(f(t)) \) based on \( f(x) = x^2 + 1 \)?
5. Can we generalize \( f(x) = x^2 + 1 \) to a higher degree polynomial?

### Tip:
When substituting expressions into a function, carefully expand the terms before simplifying to avoid common algebraic errors.

If f(x) = x^2 + 1, find and simplify the following: (a) f(t+1), (b) f(t^2 + 1), (c) f(2), (d) 2f(t), (e) (f(t))^2 + 1

Learn how to evaluate and simplify function expressions such as f(t+1), f(t^2 + 1), and more, with the function f(x) = x^2 + 1. This problem covers key algebraic concepts, including function substitution, polynomial expansion, and function evaluation. Suitable for students in Grades 9-11.

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