Find Domain, Range, Intercepts, Extrema and Asymptotes

[olyviab]

Homework Statement

f(x) = x^3/(x^2+ 1)

1. Identify Domain and Range

2. x and y intercepts (if any)

3. local and global extrema

4. Equations of the asymptotes (if any)

Homework Equations

f(x) = x^3/(x^2+ 1)

The Attempt at a Solution

f(x) = x^3/(x^2+ 1)

f'(x) = (x^2)(x^2 + 3)/(x^2+ 1)^2


1. Identify Domain and Range



2. x and y intercepts (if any)

- I found that the intercept was at x=0 and y=0 ; [0,0]

3. local and global extrema

Critical Points:
0 = x^2 + 1
x = $$\pm$$1

0 = x^2
x = 0

0 = x^2 + 3
x = $$\sqrt{-3}$$

(critical points) -1 0 1
(points to test) -2 -.5 .5 2
(increasing/decreasing) (+) (+) (+) (+)

I found that there was no extrema by doing the first derivative test.

4. Equations of the asymptotes (if any)

f(x) = x^3/(x^2+ 1)

because the degree > degree of denominator there are no horizontal asymptotes and the function spike to + infinity

derivative indicates the function is always increasing , bottom of the function is always positive , thus the top defines the behavior ---> x is all reals and y is all reals...

dividing yields f(x) = x - { x / [x² + 1 ] }---> y = x is " slant asymptote "

 

[mighty2000]

1. The domain of this function is all real numbers except x=-1, since it would result in division by zero. The range is all real numbers.

2. The x-intercept is at (0,0) and the y-intercept is also at (0,0).

3. There are no local or global extrema for this function. The first derivative test shows that the function is always increasing.

4. There are no equations for asymptotes for this function, as it does not have any vertical or horizontal asymptotes. The slant asymptote is y=x.