Here is one of the first examples I tried, taken from the 1981 high school sample exams. Consider the function f given by x -> x^2 / (x |x| + 1).

On the TI-89, it is best to enter functions in the Y= window for easy reference later. The calculator also features pretty printing of the function after entering the raw definition, as shown below using the y1 slot.

One of the questions was to solve f(x) < 1/2 x. On the TI-89, this can be done with the solve function on the just entered y1 function. The solution is given on one line, so a bit of scrolling is required, but it is correct.

Pretty neat, right?

On the TI-89, it is best to enter functions in the Y= window for easy reference later. The calculator also features pretty printing of the function after entering the raw definition, as shown below using the y1 slot.

Now, the graph feature plots the function, nicely accounting for the asymptotes.

Another question was to compute the area under the function on the interval [0,1]. This is easily done with the integral function, as shown below.

Pretty neat, right?