Question

x+3 2x Define f(x) for all real numbers x = 0. Is f a one-to-one function? Prove or give a counterexample. (Note that the write-up of the proof or counterexample should only have a few of sentences.) If the co-domain is all real numbers not equal to 1, is f an onto function? Why or why not? (Note this problem does not require a full proof or formal counterexample, just an explanation.)

Answer 1

$\text{a) }f(x)=\frac{x+3}{2x}\\ \text{ Let } x_1,x_2\in\mathbb{R}-\{0\}\text{ such that } f(x_1)=f(x_2)\\ \implies \frac{x_1+3}{2x_1}=\frac{x_2+3}{2x_2}\\ \implies\frac{1}{2}+\frac{3}{2x_1}=\frac{1}{2}+\frac{3}{2x_2}\\ \implies \frac{3}{2x_1}=\frac{3}{2x_2}\\ \implies x_1=x_2\implies f \text{ is one one}$

$\text{b) If the codomain of f is all real numbers not equal to 1, even then f is not an onto function because}\\ \nexists\text{ any } x\in \mathbb{R}\text{ such that } f(x)=\frac{1}{2}$