To determine how the graphs of the functions [tex]\( f(x) = \sqrt{16} \cdot x \)[/tex] and [tex]\( g(x) = \sqrt[3]{64} \cdot x \)[/tex] are related, let's follow a step-by-step process to compare the functions.
1. Simplify [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \sqrt{16} \cdot x
\][/tex]
We know that [tex]\( \sqrt{16} = 4 \)[/tex], so:
[tex]\[
f(x) = 4x
\][/tex]
2. Simplify [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \sqrt[3]{64} \cdot x
\][/tex]
We know that [tex]\( 64 = 4^3 \)[/tex], so:
[tex]\[
\sqrt[3]{64} = \sqrt[3]{4^3} = 4
\][/tex]
Therefore:
[tex]\[
g(x) = 4x
\][/tex]
3. Compare [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
After simplification, both functions become:
[tex]\[
f(x) = 4x
\][/tex]
[tex]\[
g(x) = 4x
\][/tex]
Since both functions are linear and have the same coefficient, their graphs are identical. This means they are equivalent.
Therefore, the correct answer is:
[tex]\[
\boxed{1 \text{ The functions } f(x) \text{ and } g(x) \text{ are equivalent.}}
\][/tex]
