# Finding a function based on the tangent line

I need your help with this question:

The tangent line to the function f(x) at x=1 is y=3x-2. Find f(x) (without using integrals).

I know that the derivative at x=1 should be 3, but without more information, how am I suppose to find f(x) ? This is a multiple answers question, but at least 3 answers gives a derivative of 3 when putting x=1.

Thank you.

• You need $f'(1)=3$, but also $f(1)=1$. – Yves Daoust Jun 17 '16 at 13:05
• Obviously many functions satisfy $f(1)=1),f'(1)=3$. What are the alternatives given? – almagest Jun 17 '16 at 13:06
• The question seems poorly written; the correct answer should be "there is no way to know." If indeed they want you to pick the function that satisfies $f(1)=1$ and $f'(1)=3$, then they should have written, "Which of the following functions could be a candidate for $f(x)$?" – Théophile Jun 17 '16 at 13:09
• I agree with you ! – user3275222 Jun 17 '16 at 13:13

You can't find $f(x)$ because you aren't given enough information. There are many functions that share the same tangent line at a given point.

But you can find $f(1)$ because $f(1)=\ell(1)$, where $\ell$ is the function whose graph is the given tangent line (i.e., $\ell(x)=3x-2$). That's because the tangent at $x=1$ passes through the point $(1,f(1))$.

EDIT: Sorry, I didn't see that you actually do have more information that you aren't telling us.

Note that you really have both $f'(1)=\ell'(1)=3$ and $f(1)=\ell(1)=1$. You may be able to eliminate all but one of the possible answers with this.

Can you also please give us the actual answers you have to choose from?

• But he has enough info. It is a multiple choice question. He just hasn't given it to us yet, so this is premature! – almagest Jun 17 '16 at 13:07
• @almagest: Ah, you are right. I didn't read the entire question. My bad. I edited my answer. – MPW Jun 17 '16 at 13:08
• Can e^(3*(x-1)) be the correct answer? – user3275222 Jun 17 '16 at 13:11
• @user3275222: If $f(x)=e^{3(x-1)}$, then $f(1)=e^0=1$, and $f'(1)=3e^0=3$. So yes, that is a possible candidate. – MPW Jun 17 '16 at 13:13
• Thanks, problem solved (and sorry for not writing the expression properly, not controlling latex yet). – user3275222 Jun 17 '16 at 13:17