To determine which option contains true statements about the graph of the function \( f(x) = 8\sqrt{x} - 3 \), let's analyze the function.
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Domain: The function involves the square root \( \sqrt{x} \), which is only defined for \( x \geq 0 \). Thus, the domain of \( f(x) \) is \( [0, \infty) \).
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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 \([-3, \infty)\).
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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.