Question

Find the function (f o g )(x)find the domain of (f o g)(x) express your answer in interval notation.

Answer 1

Composite Function

The composite function named

`(f\circ g)(x)`

is defined as:

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

We are given the functions:

`f(x)=\frac{1}{\sqrt[]{x}}`
`g(x)=x^2-4x`

The composite function is obtained by substituting g into f as follows:

`(f\circ g)(x)=\frac{1}{\sqrt[]{x^2-4x}\text{ }}`

We are required to find the domain of the composite function.

Since it's a rational function, the denominator cannot be 0, thus:

`\sqrt[]{x^2-4x}\text{ }\\e0`

The radicand of a square root must be non-negative:

`x^2-4x\ge0\text{ }`

But we must exclude 0 from the solution, thus the inequality to solve is:

`\begin{gathered} x^2-4x>0\text{ } \\ \text{Factoring:} \\ x(x-4)>0 \end{gathered}`

The product of x and x-4 must be positive. It can only happen when both are positive OR both are negative, thus:

x > 0

x - 4 > 0 => x > 4

The and combination of these conditions is (4,∞)

Now for the second condition:

x < 0

x - 4 < 0 => x < 4

The and combination of these conditions is (-∞,0)

The or combination of the solutions above is:

Solution: (-∞,0) U (4,∞)