Let f(x) = x3 + 3 and g(x) = x2 − 2. Find the rule for the function.
f − g

To find the rule for the function \( (f - g)(x) \), we subtract the expression for \( g(x) \) from the expression for \( f(x) \).

Given:
- \( f(x) = x^3 + 3 \)
- \( g(x) = x^2 - 2 \)

Now, subtract \( g(x) \) from \( f(x) \):
\[
(f - g)(x) = f(x) - g(x)
\]
\[
(f - g)(x) = (x^3 + 3) - (x^2 - 2)
\]

Simplifying the expression:
\[
(f - g)(x) = x^3 + 3 - x^2 + 2
\]
\[
(f - g)(x) = x^3 - x^2 + 5
\]

Thus, the rule for \( (f - g)(x) \) is:
\[
(f - g)(x) = x^3 - x^2 + 5
\]

Would you like further details on how this was solved or have any questions?

Here are some related questions:
1. What is the rule for \( f(x) + g(x) \)?
2. What happens when you multiply \( f(x) \) and \( g(x) \)?
3. How do you find the derivative of \( f(x) - g(x) \)?
4. What is the result of \( f(g(x)) \)?
5. How do the roots of \( f(x) \) and \( g(x) \) compare?

**Tip:** When subtracting two functions, always carefully distribute negative signs to avoid errors.

Let f(x) = x^3 + 3 and g(x) = x^2 − 2. Find the rule for the function f − g.

Learn how to find the rule for the function (f - g)(x), where f(x) = x^3 + 3 and g(x) = x^2 − 2. This step-by-step solution simplifies the expression using algebraic methods, suitable for students in Grades 9-12.

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