To analyze the statements regarding the function \( f(x) = 8\sqrt{x} - 3 \), we need to consider its domain and range.
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Domain: The expression \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( f(x) \) is \( [0, \infty) \).
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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 \( [-3, \infty) \).
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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:
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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 \([-3, ∞)\))
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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 \( f(x) = 8\sqrt{x} - 3 \).
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