Let h be the function defined by h left parenthesis x right parenthesis equals x squared plus 1​, where the domain is the set of real numbers.
​(a) Find ​h(4​).
​(b) Find ​h(negative 3​).
​(c) If h left parenthesis t right parenthesisequals26​, what are the possible values of​ t?
​(d) Find ​h(5.75​).



Let's go through each part step-by-step:

Given the function:  
\[ h(x) = x^2 + 1 \]

### (a) Find \( h(4) \):

Substitute \( x = 4 \) into the function:

\[
h(4) = 4^2 + 1 = 16 + 1 = 17
\]

### (b) Find \( h(-3) \):

Substitute \( x = -3 \) into the function:

\[
h(-3) = (-3)^2 + 1 = 9 + 1 = 10
\]

### (c) If \( h(t) = 26 \), find the possible values of \( t \):

We are given that \( h(t) = 26 \), which means:

\[
t^2 + 1 = 26
\]

Now solve for \( t \):

\[
t^2 = 26 - 1 = 25
\]

Taking the square root of both sides:

\[
t = \pm \sqrt{25} = \pm 5
\]

So, the possible values of \( t \) are \( t = 5 \) or \( t = -5 \).

### (d) Find \( h(5.75) \):

Substitute \( x = 5.75 \) into the function:

\[
h(5.75) = (5.75)^2 + 1
\]

Now calculate \( 5.75^2 \):

\[
5.75^2 = 33.0625
\]

Thus:

\[
h(5.75) = 33.0625 + 1 = 34.0625
\]

### Summary of Answers:
- (a) \( h(4) = 17 \)
- (b) \( h(-3) = 10 \)
- (c) \( t = 5 \) or \( t = -5 \)
- (d) \( h(5.75) = 34.0625 \)

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Do you want details on any step? Here are 5 related questions:
1. How would you describe the graph of \( h(x) = x^2 + 1 \)?
2. What are the intercepts of the function \( h(x) = x^2 + 1 \)?
3. Can you find \( h(0) \)?
4. What is the vertex of the function \( h(x) = x^2 + 1 \)?
5. What is the range of the function \( h(x) = x^2 + 1 \)?

**Tip:** When solving quadratic equations like \( t^2 = 25 \), always remember to consider both positive and negative roots!

Let h be the function defined by h(x) = x^2 + 1, where the domain is the set of real numbers. (a) Find h(4). (b) Find h(-3). (c) If h(t) = 26, what are the possible values of t? (d) Find h(5.75).

Learn how to solve problems involving the quadratic function h(x) = x^2 + 1. This guide covers finding h(4), h(-3), solving h(t) = 26, and calculating h(5.75). Understand key algebraic concepts, including solving quadratic equations, suitable for students in Grades 8-10.

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