Using SAGE math to find features of a function

Symmetry features
To check whether a function is even, we can type:
> bool(f(x) == f(-x)) 
True
This tells us that the function is even (is symmetric in the y-axis).

To check whether a function is odd, we can type:
> bool(f(x) == -f(-x))
False
This tells us that the function is not odd (an odd function is one that unchanged by rotation through pi radians around the origin). 
(Thanks to Dr Vincent Knight for help with this, via the SAGE mailing list.)

I'm not sure if there is a way to check whether a function is periodic in SAGE (like sin(x). I tried this but it didn't work:
> var('p')
> solve(sin(x) == sin(x+p), p)

Intercepts
The x-intercepts are the solutions of the equation f(x) = 0, for example, for f(x) = 1/(1 - x^2):
> f(x) = 1/(1 - x^2) 
> solve(f == 0, x)
[]
This tells us that this equation has no x-intercepts.

Sometimes we might need to specify a range to solve the function in eg.
> g(x) = 4*x^3 + 3*x^2 - 6*x + 4
> solve(g == 0, x)

[x == 1/8*(-I*sqrt(3) + 1)*9^(2/3) - 1/8*(I*sqrt(3) + 1)*(-1)^(1/3)*9^(1/3) - 1/4, x == -1/8*(-I*sqrt(3) + 1)*(-1)^(1/3)*9^(1/3) + 1/8*(I*sqrt(3) + 1)*9^(2/3) - 1/4, x == 1/4*(-1)^(1/3)*9^(1/3) - 1/4*9^(2/3) - 1/4]

This is not very helpful. We think there is a zero (x-intercept) in the range (-2, 0), so we can type:

> find_root(g == 0, -2, 0)

-1.8517081334935321

Hurray!

The y-intercept is the value f(0):
> f(0)
1
This tells us that the y-intercept is at y=1.

Intervals on which the function is positive, or negative
To find the intervals on which a function is positive, we can type for example:
> solve(f > 0, x)
[[x > - 1, x < 1]]
This tells us that the function is positive in the interval (-1, 1).

To find the intervals on which the function is negative, we type:
> solve(f < 0, x)
[[x < -1], [x > 1]]
This tells us that the function is negative for x < -1, and x > 1.


Intervals on which the function is increasing, or decreasing
We can use the derivative to find the intervals on which a function is increasing, for example:
> solve( diff(f, x) > 0, x)
[[x > 0, x < 1], [x > 1]]
This tells us that the function is increasing for the interval (0, 1), and for x > 1.


Likewise, we can find the intervals on which the function is decreasing by typing:
> solve( diff(f, x) < 0, x)
[[x < -1], [x > -1, x < 0]]
This tells us that the function is decreasing for x < -1, and the interval (-1, 0).

Stationary points
We can again use the derivative to find stationary points of a function, for example:
> solve( diff(f, x) == 0, x)
[x == 0]
This tells us that there is a stationary point at x = 0.

We can use the second derivative to tell whether the stationary point is a local minimum, local maximum, or inflection point:
> f2 = diff(f, x, 2)
> f2(0)
2
The second derivative is positive at the stationary point, so the stationary point is a local minimum. If the second derivative was negative, it would be a local maximum; and if the second derivative was zero, it would be an inflection point.

Asympotes
I found a nice discussion of using SAGE to find asymptotes. It says that you can find vertical asymptotes (if there are any) by finding the x-intercepts of the reciprocal of the function (ie. in our case by finding the x-intercepts of the function 1 - x^2):
> solve(1/f == 0, x)
[x == -1, x == 1]
This tells us that there are vertical asymptotes at x = -1, and x = 1.

To find the horizontal asymptotes (if any), we look at the limit of the function in +Infinity and -Infinity:
> limit(f, x =+infinity)
0
> limit(f, x=-infinity)
0
This tells us that there is a horizontal asymptote at y=0 as x approaches +Infinity, and as x approaches -Infinity.