If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be

Question

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be
$(A)(1,0)\hspace{1cm}(B)(-1,0)\hspace{1cm}(C)(-1,1)\hspace{1cm}(D)(1,1)$

This is a more than one options correct type question.The answer given is $(A),(B),(C)$.
I do not know what it means by "the function $f(x)=ax+b$ has its own inverse."
Does it mean that $f^{-1}(x)=f(x)?$

I am stuck here and can not proceed further.Please help me.Thanks.

Answer 1

According to the given answer, I think what you thought is right. Do the equation $$ax+b=\frac{x}{a}-\frac{b}{a}$$ and find the proper values of $a$ and $b$. If for all values of $x$ we have the above equation so $a=1/a,~~a\neq0$ and $b=\frac{-b}{a}$ . The results is $a=\pm1$ and if $a=+1$ then $b=0$ and if $a=-1$ then $b$ would be any numbers for exaple $1$.