Given two functions, \(f(x)\) and \(g(x)\), both functions can create composite functions \(fg\) and \(gf\).
Composite function \(fg\)
Composite function of \(fg\) can be done by substituting the function of g into the function of f.
One condition must be satisfied, range of function of g is subset of domain of function of f (\(R_g \subseteq D_f\))
Composite function \(gf\)
Composite function of \(gf\) can be done by substituting the function of f into the function of g.
One condition must be satisfied, range of function of f is subset of domain of function of g (\(R_f \subseteq D_g\))
Example 1
Given two functions:
\(f(x) = x^2, \: x \in R \)
\(g(x) = \sqrt{x}, \: x \geq 0 \)
(i) determine if composite function \(fg\) exists
(ii) determine the composite functions \(fg\) if it exists
Answer
(i) determine if composite function \(fg\) exists
Composite function \(fg\) is exist if \(R_g \subseteq D_f\)
\(R_g: y \geq 0 \)
\(D_f: x \in R \)
Since \(R_g\) is a subset of \(D_f\), \(R_g \subseteq D_f\), hence composite function \(fg\) exists
(ii) determine the composite functions \(fg\) if it exists
Substiitute the function g into the function f
\begin{equation*} \begin{split} fg & = (\sqrt{x})^2 \\ fg & = x \end{split} \end{equation*}
Example 2
Given two functions:
\(f(x) = 2x + 5, \: x \in R \)
\(g(x) = \ln x, \: x > 0 \)
(i) determine if composite function \(gf\) exists
(ii) determine the composite functions \(gf\) if it exists
Answer
(i) determine if composite function \(gf\) exists
Composite function \(gf\) is exist if \(R_f \subseteq D_g\)
\(R_f: y \in R \)
\(D_g: x > 0\)
Since \(R_f\) is not a subset of \(D_g\), \(R_f \nsubseteq D_g\), hence composite function \(gf\) does not exist
(ii) determine the composite functions \(gf\) if it exists
The composite function \(gf\) does not exist.