# proving several qualities of a given function

I'm really not sure regarding this question. would like your help if possible.

given the function:

$f(n) = \begin{cases} \frac{e^x-1}{x}, & \text{if$x \neq0$} \\ 1 & \text{if$x=0$} \end{cases}$

a) Prove for every $x\neq0$ $e^x(x-1)+1>0.$

b) Prove that $f$ in continuous in R.

c) Use the above to prove that $f$ is monotonously increasing in R.

d) Prove that every $c>0$ has a solution for $f(x)=c.$

What i did:

a) I used it as a function and tried to show that the limit is always positive, when $x \to\infty$ and $x\to\infty$ while showing that it doesn't hold for $x=0.$

b) Using the limits of a) to show continuity, with the exception of $x=0,$ which I didn't know how to prove continuous in regards to the other part of the function.

c) Didn't know how to prove it.

d)using the above, I tried to use the intermediate value theorem, by showing that since the function is continuous, then there most exist a $c$ so that $f(x)=c.$ I didn't know how to prove that only for $c>0.$

Please show me the right way to do it, as I'm sure that I've done too many errors, and I don't know how to do it correctly.

Thank you very much.

• thank you very much for your help. how would you approach to solving the inequality $e^x(x-1)+1>0$ for every $x \neq 0$? thank again – BeginningMath Jul 6 '17 at 20:09