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We are given the following functions:

- \( f(x) = 2x - 1 \)
- \( g(x) = 2x - 1 \), where \( x \neq -3 \)
- \( h(x) = \frac{(2x - 1)(x + 3)}{x + 3} \)

We need to determine if \( h(x) \) equals either \( f(x) \) or \( g(x) \).

### Step-by-Step Solution:

1. First, observe the expression for \( h(x) \):
   \[
   h(x) = \frac{(2x - 1)(x + 3)}{x + 3}
   \]

2. Simplify \( h(x) \). Notice that the term \( (x + 3) \) appears in both the numerator and denominator, so for \( x \neq -3 \), we can cancel \( (x + 3) \):
   \[
   h(x) = 2x - 1 \quad \text{(for} \, x \neq -3)
   \]

3. Now, we compare \( h(x) \) with \( f(x) \) and \( g(x) \). Since:
   \[
   f(x) = 2x - 1 \quad \text{and} \quad g(x) = 2x - 1 \, \text{for} \, x \neq -3
   \]
   Clearly, \( h(x) = f(x) = g(x) \) as long as \( x \neq -3 \).

### Conclusion:
Yes, \( h(x) \) equals both \( f(x) \) and \( g(x) \) for \( x \neq -3 \).

Do you want more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. What happens to \( h(x) \) at \( x = -3 \)?
2. How would you graph the functions \( f(x) \), \( g(x) \), and \( h(x) \)?
3. Can you find other functions that simplify in a similar way to \( h(x) \)?
4. How would you check for continuity in these functions at \( x = -3 \)?
5. What would happen if the denominator of \( h(x) \) were \( x - 3 \) instead of \( x + 3 \)?

**Tip:** Always check for values of \( x \) that make the denominator zero before simplifying!

Given the functions f(x) = 2x - 1, g(x) = 2x - 1, x ≠ -3, and h(x) = (2x - 1)(x + 3)/(x + 3), does h(x) equal f(x) or g(x)?

Explore how to simplify the rational function h(x) = (2x - 1)(x + 3)/(x + 3) and determine whether it equals f(x) or g(x). This solution covers key algebraic concepts like function simplification and cancellation of terms. Suitable for Grades 9-12 students.

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