Twelve Standard >> Composition of functions | Illustration-1

Forming New Functions through Domain Combination

Illustration 3: Let R be the set of real numbers. Define function f:R \(\rightarrow\)  R by f(x)=\(\mid x+5 \mid\) and function g:R \(\rightarrow\) R by g(x)=\(\mid 5x-2 \mid\), find fog(x) and gof(x).

Solution:

  As f:R\(\rightarrow\) R, 
    g:R\(\rightarrow\) R,
  fog & gof can e defined
   fog(x)=f(g(x))=\(\mid g(x)+5 \mid\)
            = \(\mid \mid 5x-2 \mid +5 \mid\)

gof(x)=g(f(x))
         =\(\mid g(x)-2 \mid\)
         =\(\mid 5\mid x+5 \mid-2 \mid\)

Illustration 4: Let R be the set of real numbers and f:R \(\rightarrow\)  R defined by f(x)=\((3-x^{3})^{\frac{1}{3}}\). Show that fof=\(I_R\)  is the identity function.

Solution:

We have to prove fof(x)=x
   fof(x)=f(f(x))
           =\([3-f(x^{3})]^{\frac{1}{3}}\)
           =\([3-(3-x^{3})^{\frac{1}{3} \times 3}]^{\frac{1}{3}}\)
            =\([3-3+x^{3}]^{\frac{1}{3}}\)
            =\((x^{3})^{\frac{1}{3}}\)
            =x

\(\Rightarrow\) fof=\(I_R\)