Composite Function
The composite function named
[tex](f\circ g)(x)[/tex]is defined as:
[tex](f\circ g)(x)=f(g(x))[/tex]We are given the functions:
[tex]f(x)=\frac{1}{\sqrt[]{x}}[/tex][tex]g(x)=x^2-4x[/tex]The composite function is obtained by substituting g into f as follows:
[tex](f\circ g)(x)=\frac{1}{\sqrt[]{x^2-4x}\text{ }}[/tex]We are required to find the domain of the composite function.
Since it's a rational function, the denominator cannot be 0, thus:
[tex]\sqrt[]{x^2-4x}\text{ }\ne0[/tex]The radicand of a square root must be non-negative:
[tex]x^2-4x\ge0\text{ }[/tex]But we must exclude 0 from the solution, thus the inequality to solve is:
[tex]\begin{gathered} x^2-4x>0\text{ } \\ \text{Factoring:} \\ x(x-4)>0 \end{gathered}[/tex]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,∞)
