Question

Consider the functions below.

f(x)=√x-1
g(x) = x³ + 1
Find each of the following, if possible. (If it is not possible, enter NONE.)
(a) fog
(b) gof
(c) (fog)(0)

Answer 1

`\sf a) f \circ g(x) = x`

`\sf b) g \circ f(x) = x`

`\sf c) (f \circ g)(0) = 0`

Explanation:

Given:

`f(x) = \sqrt[3]{x-1}`

`g(x) = x^3 +1`

To find:

a) fog

b) gof

c) (fog)(0)

Solution:

In order to find the compositions of the functions (f) and (g) and their values at 0, follow these steps:

`\textsf{ a) To find $f \circ g$, we first compute $f(g(x))$:}`

`\sf f(g(x)) = \sqrt[3]{(g(x))-1} = \sqrt[3]{(x^3 + 1) - 1} = \sqrt[3]{x^3} =x`

`\sf So, f \circ g(x) = x`

`\textsf{b) To find $g \circ f$, we compute $g(f(x))$}`

`\sf g(f(x)) = (f(x))^3 + 1 = \sqrt[3]{x-1})^3 + 1 = (x - 1) + 1 = x`

`\sf So, g \circ f(x) = x`

`\textsf{c) To find $ (f \circ g)(0)$, we substitute x = 0nto $f \circ g(x)$:}`

`\sf (f \circ g)(0) = f(g(0)) = f(0^3 + 1) = f(1) = \sqrt[3]{1 - 1} = \sqrt[3]{0} = 0`

`\textsf{So, $(f \circ g)(0) = 0$}`