1—Basic Stuff 22
- How does the function behave as you approach the ends of the domain? If the domain extends from
−∞to+∞, how does the function behave as you approach these regions?
- Is the function the sum or difference of two other much simpler functions? If so, you may find it easier
to sketch the two functions and then graphically add or subtract them. Similarly if it is a product.
- Is the function related to another by translation? The functionf(x) = (x−2)^2 is related tox^2 by
translation of 2 units. Note that it is translated to therightfromx^2. You can see why because(x−2)^2 vanishes
atx= +2.
- After all this, you will have a good idea of the shape of the function, so you can interpolate the behavior
between the points that you’ve found.
Example: Sketchf(x) =x/(a^2 −x^2 ).
−a a
- The domain for independent variable wasn’t given, so take it to be−∞< x <∞
- The pointx= 0obviously gives the valuef(0) = 0.
- The denominator becomes zero at the two pointsx=±a.
- If you replacexby−x, the denominator is unchanged, and the numerator changes sign. The function
is odd about zero.
−a a
- Whenxbecomes very large (|x| a), the denominator is mostly−x^2 , sof(x)behaves likex/(−x^2 ) =
− 1 /xfor largex. It approaches zero for largex. Moreover, whenxis positive, it approaches zero through
negative values and whenxis negative, it goes to zero through positive values.
−a a
- Near the pointx= 0, thex^2 in the denominator is much smaller than the constanta^2 (x^2 a^2 ). That
means that near this point, the functionfbehaves likex/a^2
