Property 14
Composite Function
\(f:A\to B\text{ and }g:B\to C\)
\(\begin{aligned}f(a)&=b\\g(b)&=c\\(g\circ f)(a)&=g(f(a))=g(b)=c\end{aligned}\)
\(g\circ f\) function is called the composite function of f and g.
- I identity function
\(f\circ f^{-1}=I\qquad f\circ I=f\qquad(f^{-1})^{-1}=f\)
\(f^{-1}\circ f=I\qquad I\circ f=f\qquad(f\circ g)^{-1}=g^{-1}\circ f^{-1}\)
1.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x+2\)
\(\Rightarrow (f\circ f)(3)=?\)
2.
\(f(x)=3x+1\)
\(g(x)=x-5\)
\(\Rightarrow (f\circ g)(2)=?\)
3.
\(f(x)=2x+1\)
\(g(x)=\sqrt{x+4}\)
\(\Rightarrow (f\circ g)(5)=?\)
4.
\(f(x)=x+4\)
\(g(x)=2x+5\)
\(\Rightarrow (f\circ g)(x)=?\)
5.
\(f(x)=x^2+1\)
\(g(x)=3x+1\)
\(\Rightarrow (g\circ f)(x)=?\)
6.
\(f(x+3)=3x+1\)
\(g(x-2)=4x+6\)
\(\Rightarrow (f\circ g)(2)=?\)
7.
\(f(x-2)=2x+1\)
\(g(x+1)=x\)
\(\Rightarrow (g\circ f)(3)=?\)
8.
\(f(x)=3x+1\)
\(g(x)=x-2\)
\(\Rightarrow (g\circ f^{-1})(0)=?\)
9.
\(f(x-2)=4x-5\)
\(g(x+1)=2x+1\)
\(\Rightarrow (f\circ g^{-1})(7)=?\)
10.
\(f(x)=-2x+4\)
\(g(x)=x-5\)
\(\Rightarrow (f\circ g)^{-1}(2)=?\)
11.
\(f(x)=2x-5\)
\(g(x)=x-2\)
\((g\circ f^{-1})^{-1}(2)=?\)
12.
\(f(x)=2x-5\)
\(g(x)=x+1\)
\((f\circ g^{-1})^{-1}(x)=?\)
13.
\(f^{-1}(x-2)=g(3x+1)\)
\(\Rightarrow (f\circ g)(10)=?\)
14.
\(f(x)=x+5\)
\((g^{-1}\circ f)(x)=2x+7\)
\(\Rightarrow g(11)=?\)
15.
\(f(x-2)=g^{-1}(x+5)\)
\(\Rightarrow (g\circ f)^{-1}(7)=?\)