Question

Trying to determine function of g(x).

Answer 1

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.

Answer 2

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

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} \: (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}`