To determine how the graphs of the functions [tex]\(f(x) = \sqrt{16} \, x\)[/tex] and [tex]\(g(x) = \sqrt[3]{64} \, x\)[/tex] are related, we need to compare their coefficients, as the coefficients determine the steepness of the lines.
1. Calculate the coefficient of [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = \sqrt{16} \, x
\][/tex]
We need to find [tex]\(\sqrt{16}\)[/tex]:
[tex]\[
\sqrt{16} = 4
\][/tex]
Therefore, the function [tex]\(f(x)\)[/tex] simplifies to:
[tex]\[
f(x) = 4x
\][/tex]
2. Calculate the coefficient of [tex]\(g(x)\)[/tex]:
[tex]\[
g(x) = \sqrt[3]{64} \, x
\][/tex]
We need to find [tex]\(\sqrt[3]{64}\)[/tex]:
[tex]\[
\sqrt[3]{64} \approx 3.9999999999999996
\][/tex]
Therefore, the function [tex]\(g(x)\)[/tex] simplifies to:
[tex]\[
g(x) \approx 3.9999999999999996 \, x
\][/tex]
3. Compare the coefficients of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
4.0 \quad \text{(for \(f(x)\)) and} \quad 3.9999999999999996 \quad \text{(for \(g(x)\))}
\][/tex]
Despite the slight numerical difference due to precision, mathematically they are very close, but strictly speaking:
[tex]\[
4.0 > 3.9999999999999996
\][/tex]
Since the coefficient of [tex]\(g(x)\)[/tex] is slightly less than the coefficient of [tex]\(f(x)\)[/tex], the function [tex]\(g(x)\)[/tex] does not increase at a faster rate than [tex]\(f(x)\)[/tex], nor are the functions equivalent.
4. Determine the correct relationship:
Given that the coefficient of [tex]\(f(x)\)[/tex] is slightly larger, at the outset, [tex]\(f(x)\)[/tex] would theoretically start at a proportionately higher rate compared to [tex]\(g(x)\)[/tex]. Thus, the correct interpretation is:
The function [tex]\(g(x)\)[/tex] has a greater initial value.
Hence, the answer is:
The function [tex]\(g(x)\)[/tex] has a greater initial value.
