Question

Use the table to answer the question. Option #1 Option #2 goes through the point (−1,−2) does not go through the point (−1,−2) domain is (−∞,∞) domain is (0,∞) range is (−∞,∞) range is (0,∞) Which option contains true statements about the graph of f(x)=8x−−√3 ? (1 point) Option #

Answer 1

To determine which option contains true statements about the graph of the function \( f(x) = 8\sqrt{x} - 3 \), let's analyze the function.

1. **Domain**: The function involves the square root \( \sqrt{x} \), which is only defined for \( x \geq 0 \). Thus, the domain of \( f(x) \) is \( [0, \infty) \).

2. **Range**: To find the range, consider the behavior of \( f(x) \) as \( x \) varies over its domain. When \( x = 0 \):
\[
f(0) = 8\sqrt{0} - 3 = -3.
\]
As \( x \) increases, \( \sqrt{x} \) increases and consequently \( f(x) \) will continue to increase without bound. Therefore, the range starts from \(-3\) and goes to \(\infty\), which can be expressed as \([-3, \infty)\).

3. **Point**: The function \( f(x) = 8\sqrt{x} - 3 \) is defined for \( x = 0 \) and does not go through the point \((-1, -2)\) since \(-1\) is outside the domain.

Based on this analysis:

- **Domain** is \((0, \infty)\): **True**.
- **Range** is \((0, \infty)\): **False** (it's \([-3, \infty)\)).
- **Goes through the point (-1, -2)**: **False** (it only goes through points in its domain).
- **Does not go through the point (-1, -2)**: **True**.

### Conclusion
To answer your question, Option #1 contains the true statements about the graph of \( f(x) = 8\sqrt{x} - 3\) based on the function's domain and range analysis. Therefore, **Option #1 is correct**.