Mathematics - Functions Question with Solution | TestHub

We have, $\mathrm{f}(\mathrm{x})=\mathrm{x}+(-1{)}^{\mathrm{x}-1}$.

Putting successively $\mathrm{x}=1,2,3,\dots$, we get

$$\begin{array}{rlr}& f(1)=1+1=2;f(2)=2+(-1)=1;f(3)=3+(-1{)}^2=3+1=4;f(4)-1=3& \end{array}$$

$$f(4)=5+1=6;f(5)=6-1=5;\dots .$$

From above, we see that the graph of $f(x)$ consists of the points $(1,2),(2,1),(3,4),(4,3)$, $(5,6),(6,5),\dots$. Thus if $(\mathrm{a},\mathrm{b})$ is a point on the graph, then $(\mathrm{b},\mathrm{a})$ is also a point on the graph. Hence the inverse of is f itself i.e. $f^{-1}(x)=x+(-1{)}^{x-1};x\in N$.