# Analytically determine if $f(x) = f'(x)$ is possible?

I was taking a test and two true/false type questions were asked.

In one of them, I had to say if there is a function $$f(x)$$ such that $$f(x) = f'(x)$$. Of course, $$e^x$$ is such a function and almost everyone who has taken a calculus course knows this fact well.

In the other question, I had to determine if $$f(x) = -f'(x)$$ was possible.

I was completely stumped at this one. I had never before encountered a function with such property nor did I know how to approach this problem analytically as I am just a high school student.

My question is: is there an analytical way to determine if such a function exists? By analytical, I mean no guessing allowed and just giving an example won't be enough.

Is this possible? If not, can you give an example of a function with the above property?

• Also, $f(x)=0$. – Micapps Feb 4 '19 at 9:40
• If you know the formula $\frac d {dx} [f(-x)]=-[\frac d {dx} f](-x)$ then there is not much to guess. – Kavi Rama Murthy Feb 4 '19 at 9:41

If $$f'(x)=f(x)$$ and if $$g(x)=f(-x)$$, then $$g'(x)=-f'(-x)=-f(-x)=-g(x)$$. Can you take it from here?
Linear differential equations $$ay"+by'+cy$$ are solved by considering the roots of the polynomial $$ax^2+bx+c$$, here you have $$x+1=0$$.
As you're in high school, you have probably not covered the topic of differential equations but you can use one to find the analytical solution to your question. $$dy/dx + y = 0$$ and you will find that the solution is $$y=ce^{-x}$$ (where c is any constant): https://www.wolframalpha.com/input/?i=dy%2Fdx+%2B+y+%3D+0