Showing that $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 4x - 5$ is onto

Question

Hello attached is my attempt of the problem. I hope I did this correctly. Does this make sense? Thank you for your help.

Answer 1

You can show it in two ways.

The first is your method, for every $y \in \mathbb{R}$ we can find $x\in \mathbb{R}$ such that $y=f(x)$ that is

$$x=\frac{y+5}{4}$$

The second is to observe that $f(x)$ is continuos and

$$\lim_{x\to +\infty} f(x)=+\infty$$

$$\lim_{x\to -\infty} f(x)=-\infty$$

thus for IVT for each $y$ we can find $x$ such that $y=f(x)$.