I got a question on whether the function is one-one or many-one and onto or into .

Question

The question is: Let $f\colon \mathbb{R} \longrightarrow \mathbb{R}$ be a function defined by $f(x)= \frac{x-m}{x-n}$ where $m \not = n$.

1) is it one-one or many-one

2) is it into or onto

I tried and I think my answer is one one and into but I don't think I followed the right steps to . if someone could show me how to solve the 1st part with concept of differentiability ( strictly increasing or strictly decreasing) and also the second part. I couldn't find the solution to this on the web so here I am. Thanks .

Answer 1

HINT (for $f(x)=\frac{x-m}{x-n}$)

$$ \frac{x-m}{x-n}=\frac{x-n+n-m}{x-n}=1+\frac{n-m}{x-n}. $$ Can you see the shape of the graph of it now?

Answer 2

(a) f is one-to-one (injective) if f maps every element of A to a unique element in B. In other words no element of B are mapped to by two or more elements of A ,mathematically saying if $f(x_{1})=f(x_{2})\implies x_{1}=x_{2}$

if $\dfrac{x_{1}-m}{x_{1}-n}=\dfrac{x_{2}-m}{x_{2}-n}$

$x_{1}(m-n)=x_{2}(m-n)$

$(x_{1}-x_{2})(m-n)=0$ and $m\neq n$

so, $x_{1}=x_{2}$

so this function is one one as you found yourself

(b)f is onto (surjective)if every element of B is mapped to by some element of A. In other words, nothing is left out.

$y=\dfrac{x-m}{x-n}\implies$ $x=\dfrac{ny-m}{y-1}$

corresponding to $y=f(x)=1$ there is no $x$ present in it's domain so function is not surjective.