Mathematics - Functions Question with Solution | TestHub

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

Putting successively $x=1,2,3,\dots$, we get:

$$\begin{array}{rl}f(1)& =1+1=2\\ f(2)& =2+(-1)=1\\ f(3)& =3+(-1{)}^2=3+1=4\\ f(4)& =4+(-1{)}^3=4-1=3\\ f(5)& =5+(-1{)}^4=5+1=6\\ f(6)& =6+(-1{)}^5=6-1=5\\ & \dots \end{array}$$

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