Question

How are the graphs of the functions f(x) = √16^x and g(x) =3√64^x related?

Answer 1

Answer:- The functions f(x) and g(x) are equivalent.

Explanation:-

Given functions:- $f(x)=\sqrt{16^x}$ and $g(x)=\sqrt[3]{64^x }$

Simplify the functions by using law of exponents

$(a^n)^m=a^{nm}$

Thus $f(x)=\sqrt{16^x}=\sqrt{(4^2)^x}=\sqrt{4^{2x}} =\sqrt{(4^x)^2}=4^x$

and $g(x)=\sqrt[3]{64^x}=\sqrt[3]{(4^3)^x}= \sqrt[3]{4^{3x}}= \sqrt[3]{(4^x)^3}=4^x$

⇒f(x)=g(x)

Therefore ,the functions f(x) and g(x) are equivalent.

Answer 2

Both are exponential function & increasing

The function g(x) increases at a faster rate: Proof
For x=0,  f(x) =1
for x=0, g(x) =3