Question

Trying to determine function of g(x).

Answer 1

Answer:

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

Step-by-step explanation:

We can see that the curve has not been translated horizontally or vertically, nor has it been reflected in either axis.  Therefore, g(x) is a stretch:

$y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis by a factor of}\:a$

$y=f(a x) \implies f(x) \: \textsf{stretched parallel to the x-axis by a factor of} \: \dfrac{1}{a}$

Parent function $f(x)=\sqrt[3]{x}$

$\implies g(x)=a\:f(x)=a\sqrt[3]{x}$

$\textsf{or}\quad g(x)=f(ax)=\sqrt[3]{ax}$

From inspection of the graph:

$g(1)=4$

Therefore:

$\implies a\sqrt[3]{1}=4$

$\implies a=4$

$\implies g(x)=4\sqrt[3]{x}$

Or:

$\implies \sqrt[3]{a(1)}=4$

$\implies a=4^3$

$\implies a=64$

$\implies g(x)=\sqrt[3]{64x}$

$\implies g(x)=\sqrt[3]{64}\sqrt[3]{x}$

$\implies g(x)=4\sqrt[3]{x}$

Therefore, the translated function is:

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

Answer 2

Answer:

To find f(g(x)), we just substitute x = g(x) in the function f(x). For example, when f(x) = x2 and g(x) = 3x - 5, then f(g(x)) = f(3x - 5) = (3x - 5)2.