CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Let $$f=\left\{ \left( 3,1 \right) ,\left( 9,3 \right) ,\left( 12,4 \right)  \right\} $$ and $$g=\left\{ \left( 1,3 \right) ,\left( 3,3 \right) ,\left( 4,9 \right) ,\left( 5,9 \right)  \right\} $$. Show that $$g\circ f$$ and $$f\circ  g$$ are both defined. Also, find $$f\circ g$$ and $$g\circ f$$.


Solution

$$f\circ g$$ means $$g(x)$$ function is in $$f(x)$$ function. 
$$g\circ f$$ means $$f(x)$$ function is in $$g(x)$$ function.
$$f=\{(3,1),(9,3),(12,4)\}$$ and $$g=\{(1,3),(3,3),(4,9),(5,9)\}$$
$$f=\{3,9,12\}\rightarrow \{1,3,4\}$$ and $$g:\{1,3,4,5\}\rightarrow \{3,9\}$$

Co-domain of $$f$$ is a subset of the domain of $$g.$$
So, $$g\circ f$$ exists and $$g\circ f:\{3,9,12\}\rightarrow \{3,9\}$$
$$\Rightarrow$$  $$(g\circ f)(3)=g[f(3)]=g(1)=3$$
$$\Rightarrow$$  $$(g\circ f)(9)=g[f(9)]=g(9)=3$$
$$\Rightarrow$$  $$(g\circ f)(12)=g[f(12)]=g(4)=9$$
$$\Rightarrow$$  $$g\circ f=\{(3,3),(9,3),(12,9)\}$$

Co-domain of $$g$$ is a subset of the domain of $$f$$.
So, $$f\circ g$$ exists and $$f\circ g:\{1,3,4,5\}\rightarrow \{3,9,12\}$$
$$\Rightarrow$$  $$(f\circ g)(1)=f[g(3)]=f(3)=1$$
$$\Rightarrow$$  $$(f\circ g)(3)=f[g(3)]=f(3)=1$$
$$\Rightarrow$$  $$(f\circ g)(4)=f[g(4)]=f(9)=3$$
$$\Rightarrow$$  $$(f\circ g)(5)=f[g(5)]=f(9)=3$$
$$\Rightarrow$$  $$f\circ g=\{(1,1),(3,1),(4,3),(5,3)\}$$

Mathematics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image