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 analyze the statements regarding the function \( f(x) = 8\sqrt{x} - 3 \), we need to consider its domain and range.

1. **Domain**: The expression \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( f(x) \) is \( [0, \infty) \).

2. **Range**: As \( x \) starts from 0 and increases toward infinity, \( \sqrt{x} \) starts from 0 and increases toward infinity. Thus, \( f(0) = 8\sqrt{0} - 3 = -3 \). As \( x \) approaches infinity, \( f(x) \) approaches infinity. Therefore, the range of the function is \( [-3, \infty) \).

3. **Point \((-1, -2)\)**: The function \( f(x) \) is only defined for \( x \geq 0 \), so it does not pass through the point \((-1, -2)\) because \(-1\) is outside of the domain.

Based on this analysis:

- **Option #1**:
- Goes through the point \((-1, -2)\): **False** (it does not go through this point)
- Domain is \((-∞, ∞)\): **False** (the domain is \([0, ∞)\))
- Range is \((-∞, ∞)\): **False** (the range is \([-3, ∞)\))

- **Option #2**:
- Does not go through the point \((-1, -2)\): **True**
- Domain is \((0, ∞)\): **False** (the domain is actually \([0, ∞)\))
- Range is \((0, ∞)\): **False** (the range is \([-3, ∞)\))

None of the statements in either option are wholly true, but Option #2 contains the only true statement about not going through the point \((-1, -2)\).

Thus, the answer is:
**Option #2** contains the true statement about the graph of \( f(x) = 8\sqrt{x} - 3 \).