Consider the following function. f(x) = x^2 / x^2 − 25 .Complete the following table below. (Round your answers to two decimal places.)

Question

x	−5.5	−5.1	−5.01	−5.001	−5	−4.999	−4.99	−4.9	−4.5
f(x)					?

2.Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary. If an answer does not exist, enter DNE.)

h(x) = x2 − 81 / x3 + 9x2 − x − 9

 (smaller value) =

 

(larger value) =

Answer 1

For problem 1), just plug in the x-value into the function. For instance for x = =5.5, f(-5.5) = (-5.5)²/[(-5.5)² - 25] = 30.25/(30.25 - 25) = 30.25/5.25 = 5.76

Of course, when you get to -5, you will no longer be able to plug in the x-value since that would be dividing be zero, So that function value does not exist.

For problem 2), you must factor the numerator and denominator. Factor the numerator as the difference of two squares (a² - b²) = (a + b)(a - b). Factor the denominator by grouping:

x³ + 9x² - x - 9 = (x³ + 9x²) + (- x - 9) = x²(x + 9) + -1(x + 9) = (x + 9)(x² - 1) then factor "x² - 1" as the difference of two squares as you did the numerator.

When factored, you can cancel like-terms on the top and bottom. After that, what is left in the denominator will define the asymptotes. Since the denominator can not be zero, the asymptotes will be the result of making the denominator equal to zero. For instance, if one of the factors of the denominator, after canceling, was "x + 1" then if you set "x + 1" equal to zero you will get x = -1 as a vertical asymptote.