As a follow-up to Understanding why $f(x)=2x$ is injective, I'm working on proving/disproving that $$f(x)=3x+4,$$ where inputs/outputs live on real numbers, is injective and surjective.
Supposing that $$f(a)=f(b),$$ then $$3a+4=3b+4.$$
Solve for $0$:
So, $a$ must equal $b$. Therefore, $f$ is injective.
With respect to whether it's surjective, I looked at its graph:
Since $$3x + 4$$ is linear, then it's continuous, I believe. As a result, is that proof enough that it's surjective, i.e. for all $x$ in $$f(x),$$ the output will cover all real numbers?