Composite Function

Basic Concept

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.

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