Do all straight lines have an inverse function?

Question

Do all straight lines have an inverse function? It would seem to make sense. All linear lines would pass the horizontal line test and thus when reflected across $y=x$ it would still be a function. However, the answers says no, and I cannot find a case where my logic doesn't work. Could I have some insight please.

Answer 1

Characteristic for non-vertical "straight lines" is that they correspond with functions that can be prescribed by $x\mapsto ax+b$ where $a,b$ are fixed real numbers.

Based on equation $y=ax+b$ we can find an expression for $x$ in $y$ under the extra condition that $a\neq0$: $$x=\frac1{a}(y-b)$$By interchanging $x$ and $y$ we find the inverse function is:$$y=\frac1{a}(x-b)$$ This tells us that such linear functions have an inverse if $a\neq0$. In case $a=0$ we are dealing with a constant function prescribed by $x\mapsto b$. This function is not injective hence has no inverse.

Answer 2

1) Answer is no:

$X- $axis: $y = f(x) = 0$, $ x \in \mathbb{R} $, and

$Y-$axis : $x= g(y) = 0$, $y \in \mathbb{R}$

do not have inverses.

2) Let $y = f(x) = mx + c$, with $m \ne 0$, $m$ real.

Then $y' = f'(x) = m $;

A) $m\gt 0$: $f'(x) = m \gt 0$, i.e.

$f(x)$ is strictly mon. increasing everywhere.

B) $m\lt 0$: $f'(x) = m \lt 0$, i.e.

$f(x)$ is strictly mon. decreasing everywhere.

$f(x)$ in case A) or B) has an inverse functions.