How can i prove this statement?

Question

Statement: Let 𝑓:[𝑎, 𝑏] → [𝑐, 𝑑] for 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑅 be continuous on (𝑎, 𝑏). Then for any point 𝑞 ∈ (𝑐, 𝑑) there exists a point 𝑝 ∈ (a, b) such that 𝑓(𝑝) = 𝑞.

I know that I need to use intermediate value theorem, but can't seem to find a way.

Answer 1

The IVT states that if f is continuous on [a,b] then it takes on all values between f(a) and f(b) over that interval.

So we need to have f(a) = c, and f(b) = d to prove your statement.

Answer 2

You answered your own question!

Presumably c and d are the maximum and minimum values in the range of f on [a,b], although it is not clear from the statement of the problem.

The intermediate value theorem says that f takes on every value between the maximum and minimum in the range of f, which is equivalent to the statement to be proved.