https://file.aiquickdraw.com/mathbot/file/1726620992774jek7k0qy.jpg

The function provided in the image is \( f(x) = -12 + \sqrt{x + 33} \). We need to calculate the following values: \( f(-4) \), \( f(-1) \), \( f(0) \), and \( f(3) \).

Let's calculate each one step by step.

### 1. \( f(-4) \)
Substitute \( x = -4 \) into the function:
\[
f(-4) = -12 + \sqrt{-4 + 33} = -12 + \sqrt{29}
\]
\[
f(-4) = -12 + 5.385 \approx -6.615
\]

### 2. \( f(-1) \)
Substitute \( x = -1 \) into the function:
\[
f(-1) = -12 + \sqrt{-1 + 33} = -12 + \sqrt{32}
\]
\[
f(-1) = -12 + 5.657 \approx -6.343
\]

### 3. \( f(0) \)
Substitute \( x = 0 \) into the function:
\[
f(0) = -12 + \sqrt{0 + 33} = -12 + \sqrt{33}
\]
\[
f(0) = -12 + 5.745 \approx -6.255
\]

### 4. \( f(3) \)
Substitute \( x = 3 \) into the function:
\[
f(3) = -12 + \sqrt{3 + 33} = -12 + \sqrt{36}
\]
\[
f(3) = -12 + 6 = -6
\]

### Summary of results:
- \( f(-4) \approx -6.615 \)
- \( f(-1) \approx -6.343 \)
- \( f(0) \approx -6.255 \)
- \( f(3) = -6 \)

Let me know if you need further details or explanations.

Here are 5 questions to expand this:
1. How do you handle negative inputs inside a square root?
2. How does changing the constant inside the square root affect the function?
3. What is the domain of the function \( f(x) = -12 + \sqrt{x + 33} \)?
4. How would you find the range of this function?
5. What transformations are applied to the parent function \( f(x) = \sqrt{x} \)?

**Tip:** Always ensure the value under the square root is non-negative to maintain real-valued results.

Given the function f(x) = -12 + sqrt(x + 33), calculate f(-4), f(-1), f(0), and f(3).

Calculate specific function values for f(x) = -12 + sqrt(x + 33), including f(-4), f(-1), f(0), and f(3). Understand key algebraic concepts such as substitution and square roots. This problem is suited for students in Grades 9-10.

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