Question

What are the domain and range of f(x) = 3(x+5)2 − 8?
Responses
A Domain: All real numbers
Range: x ≥ 5Domain: All real numbers Range: x ≥ 5
B Domain: x ≥ 5
Range: f(x) ≥ −8Domain: x ≥ 5 Range: f(x) ≥ −8
C Domain: f(x) ≤ −8
Range: x ≥ 5Domain: f(x) ≤ −8 Range: x ≥ 5
D Domain: All real numbers
Range: f(x) ≥ −8

Answer 1

To determine the domain and range of the function \( f(x) = 3(x+5)^2 - 8 \), let's analyze it step by step.

1. **Domain**:

The function \( f(x) = 3(x+5)^2 - 8 \) is a polynomial function (specifically, a quadratic function). Polynomial functions are defined for all real numbers. Therefore, the domain is:
- **Domain**: All real numbers.

2. **Range**:

Next, we find the range. The function can be rewritten to identify its vertex form:
- The term \( (x+5)^2 \) will always be non-negative (i.e., it is zero or positive).
- Therefore, the minimum value of \( (x+5)^2 \) occurs when \( x = -5 \), giving a value of 0.

Now substituting this minimum value back into the function:
\[
f(-5) = 3(0) - 8 = -8
\]
Since the coefficient of the quadratic term (3) is positive, the parabola opens upward, indicating that the minimum value of the function is \( -8 \). Therefore, the range is:
- **Range**: \( f(x) \geq -8 \).

Combining both analyses, the final answer is:
- **Domain**: All real numbers.
- **Range**: \( f(x) \geq -8 \).

Thus, the correct response is:
**D: Domain: All real numbers; Range: f(x) ≥ -8.**