# I need help with an assignment question please for numerical methods

Consider the function $f (x) = xe^x - 2,$ we want to study the properties of $f (x)$ so that we can apply numerical methods to solve the equation $f (x) = 0$.

Which option is false ?

1. the function, $f (x)$ is well defined and continuous for all $x$ in the interval $(0,2)$

2. the function, $f (x)$ has no discontinuity and no singularities

3. the function, $f '(x)$ is well defined and continuous for all $x$ in the interval $(0,2)$

4. the function, $f '(x)$ has no discontinuity and no singularities

5. all of the above

I do not understand the different options given. How do I know if the function is well defined?

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"Well-defined", in this context, just means, if you plug in a value of $x$, you get a value of $f$. – Gerry Myerson Jul 15 '13 at 12:39
Given any $x\in (0,2)$, is there a number $f(x)$ defined without ambiguity? If so, the function is well-defined on $(0,2)$. – 1015 Jul 15 '13 at 12:39

Hint 1: Here is a plot of $f(x)$ and $f'(x)$ over the range (do they look continuous over the range, see any singularities over $\mathbb{R}$). Can you make an analytical argument over the range $(0,2)$ and $\mathbb{R}$, respectively, regarding continuity and singularities? Are both continuous? Do either of them have singularities?
Hint 2: Here is a plot of $f(x) = 0$.
Hint 3: From hint 2, did you try solving $f(x) = 0$ analytically? Can you?
What if you had $1/(x-1)$. Where is the discontinuity? Where is the singularity (en.wikipedia.org/wiki/Mathematical_singularity)? – Amzoti Jul 15 '13 at 13:50