Answer :
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]
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]
