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$$f(x)= \frac{x}{(1-x)^2}$$

I have been trying to sketch functions with a repeated root in the denominator. However, I cannot do it as I struggle to find where $x$ intersects the graph and the shape of it as I cannot split it into partial fractions. I am unsure whether there is another method of doing these particular ones.

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  • $\begingroup$ To find $x$-intercepts, if that's what you ask: A fraction is $0$ if and only if its numerator is $0$ and its denominator is not. $\endgroup$
    – Torsten Schoeneberg
    Apr 18, 2019 at 22:57
  • $\begingroup$ do you know how you would find range and domain of this function as I kind of understand the sketching but now I am stuck on this $\endgroup$
    – user221435
    Apr 18, 2019 at 23:07
  • $\begingroup$ For domain look where the function is defined! So when denominator equals to zero the function isn't defined so $(1-x)^2=0$ is not in the domain. By which the answer to that equation! $\endgroup$
    – user635953
    Apr 18, 2019 at 23:12
  • $\begingroup$ Note that you can make a plot up to $x=1$ (or just under $1$) and another plot for $x=1$ (or just above $1$) onward. The split at $1$ is because it is a root of the denominator. $\endgroup$
    – Ertxiem - reinstate Monica
    Apr 18, 2019 at 23:16
  • $\begingroup$ The range for this equation is f(x)<1/4. I am unsure why it is this and where they got 1/4 from. $\endgroup$
    – user221435
    Apr 18, 2019 at 23:21

1 Answer 1

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A good method I use to plot graphs:

A) Domain

B) Intercepts $x,y=0?$

C) Symmetry/Periodicity:

Symmetry: If $f(-x)=f(x)$ on the domain then it is EVEN (symmetric about y axis).

Or $f(-x)=-f(x)$ on domain then it is ODD (symmetric about the origin).

Can be neither odd or even.

Periodicity: Where $f(x+p)=f(x)$ where $p$ is a positive constant.

D) Asymptotes (horizontal/vertical)

E) Intervals of increase or decrease ($f'(x)$)

F) Local Min/Max or Inflection ($f'(x)=0$)

G) Concavity ($f''(x)$)

For your case; $x=0$, then $y=0$

All these will help you determine shape, you do not necessarily need partial fractions!

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  • $\begingroup$ Hello, what would you do to check symmetry and periodicity. $\endgroup$
    – user221435
    Apr 18, 2019 at 23:06
  • $\begingroup$ I will edit my answer! $\endgroup$
    – user635953
    Apr 18, 2019 at 23:07

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