Twelve Standard >> Composition of a function | Inverse function example problems

Examples of Inverse function

Example: 
Let f:N \(\rightarrow\) Y be a function as f(x)=2x-5, where codomain Y={y:y=2x-5, x \(\in\)}. Show that f is invertible. Also find the inverse of f.

Solve:
Let \(x_1\), \(x_2\) \(\in\) N such that
     f\(x_1\)= f\(x_2\)
 \(\therefore\) 2\(x_1\)-5=2\(x_2\)-5
        [\(\because\) f(x)=2x-5]
 \(\therefore\) 2\(x_1\)=2\(x_2\)
 \(\therefore\) \(x_1\)=\(x_2\)
 \(\therefore\) f is one-one (Proved)

y \(\in\) Y, y=f(x)
           \(\therefore\) y=2x-5
           \(\therefore\) x=\(\frac{y+5}{2}\)
When y=-1, x=\(\frac{-3+5}{2}\)
                      =\(\frac{2}{2}\)=1

For all y \(\in\) Y, there xist a x \(\in\) N. Such that 
f(x)=2x-5=2\(\frac{y+5}{2}\)-5
               =y+5-5=y
\(\therefore\) f is Onto 
\(\therefore\) f is a bisective & invertible.

Now, invers of f
                      =\(f^{-1}(y)\)
                      =\(\frac{y+5}{2}\)
                      is the inverse of f

Alternatively method:

Consider any arbitary element y \(\in\) Y, by the defination of Y, y=2x-5 for all x\(\in\) N so that
  y=2x-5
\(\therefore\) x=\(\frac{y+5}{2}\)
                 =g(y)
Since g:y \(\rightarrow\) N by g(y)= \(\frac{y+5}{2}\)
gof(x)=g(f(x))
         =\(\frac{f(x)+5}{2}\)
         =\(\frac{2x-5+5}{2}\)
         =x
 fog(y) = f(g(x))=2g(x)-5
          =2\(\times \frac{y+5}{2}\)-5
          =y+5-5
          =y

  gof(x)=x
\(\Rightarrow\) gof=\(I_N\)
  fog(y)=y
\(\Rightarrow\) fog=\(I_Y\)
So, g=\(f^{-1}\)