Question

If h(x)=(f o g) (x) and h(x)= ∛x+3, find g( x) if f(x)= ∛x+2 (picture attached)

Answer 1

Answer:

g(x) = x+1

Step-by-step explanation:

Informally, you can see that the function h(x) takes the root of a value that is 1 more than the value under the same radical in f(x). This suggests that adding 1 to x in f(x) will give you h(x). That is, ...

h(x) = f(x+1) = f(g(x))

so

g(x) = x+1

_____

More formally, you can apply the inverse of the function f(x) to the equation ...

h(x) = f(g(x))

f^-1(h(x)) = f^-1(f(g(x))) . . . inverse function applied

f^-1(h(x)) = g(x) . . . . . . . . . simplified

Now f^-1(x) can be found by solving for y in ...

x = f(y)

x = ∛(y+2) . . . . . . . . . definition of f(y)

x^3 = y+2 . . . . . . . . . cube both sides

x^3 -2 = y = f^-1(x) . . . subtract 2 from both sides

So, f^-1(h(x)) is ...

f^-1(h(x)) = g(x) = (∛(x+3))^3 -2 = x+3 -2

g(x) = x+1