Can I use Intermediate Value Theorem this way?

Question

The question says:

Prove that $f(x) = 3$ has a solution on the interval $[a,b]$

And Intermediate Value Theorem says that

if $f(a) * f(b) < 0$, then it has a solution on that interval

So instead of seeing if $3$ is between the interval and stuff like that. Can't I just do this:

$$f(x) = 3$$ $$f(x) - 3 = 0$$

And then we consider $f(x) - 3$ a completely new whole function called $g(x) = 0$

According to the theorem, I can say that since $g(a) * g(b) < 0$, it has a solution on the interval.

Can I do that?

Answer 1

Yes, you can do this. There is a variant of this argument which allows you to prove the Brouwer fixed point theorem in one dimension: suppose $f:[a,b]\to [a,b]$ is a continuous function. Show there exists a $c\in [a,b]$ with $f(c)=c$.

It's a great exercise to try out, and from your question you are halfway there already.