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The problem asks you to evaluate the function \( f(x) = x^2 + 3x + 7 \) for three different values: \( f(-3) \), \( f(x+1) \), and \( f(-x) \). Let's go through each one step by step:

### a. \( f(-3) \)
Substitute \( x = -3 \) into the function:

\[
f(-3) = (-3)^2 + 3(-3) + 7 = 9 - 9 + 7 = 7
\]

So, \( f(-3) = 7 \).

### b. \( f(x+1) \)
Substitute \( x+1 \) into the function:

\[
f(x+1) = (x+1)^2 + 3(x+1) + 7
\]

Expanding the squares and simplifying:

\[
(x+1)^2 = x^2 + 2x + 1
\]
\[
3(x+1) = 3x + 3
\]

Now substitute into the function:

\[
f(x+1) = (x^2 + 2x + 1) + (3x + 3) + 7 = x^2 + 5x + 11
\]

So, \( f(x+1) = x^2 + 5x + 11 \).

### c. \( f(-x) \)
Substitute \( -x \) into the function:

\[
f(-x) = (-x)^2 + 3(-x) + 7
\]

Simplify:

\[
(-x)^2 = x^2
\]
\[
3(-x) = -3x
\]

Now substitute into the function:

\[
f(-x) = x^2 - 3x + 7
\]

So, \( f(-x) = x^2 - 3x + 7 \).

### Final answers:
- \( f(-3) = 7 \)
- \( f(x+1) = x^2 + 5x + 11 \)
- \( f(-x) = x^2 - 3x + 7 \)

Would you like further details on any of these steps?

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Here are 5 follow-up questions to deepen your understanding:
1. How would you evaluate \( f(x) \) for another arbitrary value, such as \( f(2) \)?
2. What happens to the function \( f(x) \) if you substitute \( x = 0 \)?
3. How would the result change for \( f(x-1) \)?
4. Can you derive the vertex form of the quadratic function \( f(x) = x^2 + 3x + 7 \)?
5. How would you find the roots of \( f(x) = 0 \)?

**Tip:** Always simplify expressions step by step to avoid mistakes in expanding or factoring terms.

Evaluate the function f(x) = x^2 + 3x + 7 at the given values of the independent variable: a. f(-3), b. f(x+1), c. f(-x).

This page covers how to evaluate the quadratic function f(x) = x^2 + 3x + 7 at different values, including f(-3), f(x+1), and f(-x). Learn step-by-step how to substitute values into the function and simplify.

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