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We are given the function \( g(x) = 3x^2 - 2x + 7 \), and we are asked to find the values of \( g(x) \) for different inputs.

Let's solve each part step by step:

### a) \( g(0) \)
Substitute \( x = 0 \) into the function:
\[
g(0) = 3(0)^2 - 2(0) + 7 = 0 - 0 + 7 = 7
\]

### b) \( g(-2) \)
Substitute \( x = -2 \) into the function:
\[
g(-2) = 3(-2)^2 - 2(-2) + 7 = 3(4) + 4 + 7 = 12 + 4 + 7 = 23
\]

### c) \( g(3) \)
Substitute \( x = 3 \) into the function:
\[
g(3) = 3(3)^2 - 2(3) + 7 = 3(9) - 6 + 7 = 27 - 6 + 7 = 28
\]

### d) \( g(-x) \)
Substitute \( x = -x \) into the function:
\[
g(-x) = 3(-x)^2 - 2(-x) + 7 = 3x^2 + 2x + 7
\]

### e) \( g(1 - t) \)
Substitute \( x = 1 - t \) into the function:
\[
g(1 - t) = 3(1 - t)^2 - 2(1 - t) + 7
\]
First, expand \( (1 - t)^2 \):
\[
(1 - t)^2 = 1 - 2t + t^2
\]
Now substitute:
\[
g(1 - t) = 3(1 - 2t + t^2) - 2(1 - t) + 7
\]
Distribute:
\[
g(1 - t) = 3(1) - 3(2t) + 3(t^2) - 2(1) + 2t + 7
\]
Simplify:
\[
g(1 - t) = 3 - 6t + 3t^2 - 2 + 2t + 7 = 3t^2 - 4t + 8
\]

### Final answers:
- a) \( g(0) = 7 \)
- b) \( g(-2) = 23 \)
- c) \( g(3) = 28 \)
- d) \( g(-x) = 3x^2 + 2x + 7 \)
- e) \( g(1 - t) = 3t^2 - 4t + 8 \)

Let me know if you'd like further details or have any questions!

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Here are some related questions:
1. What is the general formula for finding the vertex of a quadratic function?
2. How would you compute the derivative of \( g(x) = 3x^2 - 2x + 7 \)?
3. Can the function \( g(x) \) have any real roots? How would you determine that?
4. How would the graph of \( g(x) = 3x^2 - 2x + 7 \) look like?
5. What is the value of \( g(5) \)?

**Tip:** When expanding squared binomials like \( (1 - t)^2 \), remember to apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).

Given that g(x) = 3x^2 - 2x + 7, find g(0), g(-2), g(3), g(-x), and g(1 - t).

Learn how to solve the quadratic function g(x) = 3x^2 - 2x + 7 for different inputs including g(0), g(-2), g(3), g(-x), and g(1 - t). This problem explores key algebraic concepts and substitution methods for quadratic functions, suitable for high school students in grades 9-11.

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