Algebraic Function

Question

Which of the following statements is (are) true when 𝒇(𝒙) = 𝒙𝟐−𝟐𝒙−𝟑/𝒙𝟑+𝟑𝒙𝟐−𝟔𝒙−𝟖?

I. The graph 𝑓(𝑥) has two vertical asymptotes at 𝑥 = 2 and 𝑥 = −4.

II. The 𝑥- and 𝑦-intercepts of the graph of 𝑓(𝑥) are both 3.

a. I

b. II

c. I and II

d. Neither statement is true.

Answer 1

I. is true since both 2 and -4 make the denominator equal to zero

II. is false since f(0) = 0.375

Answer 2

To analyze for vertical asymptotes, factor the top and bottom: .

The x+1 on numerator and denominator will divide out and simplify the expression to .

The vertical asymptotes will be where the factors on the denominator are equal to zero: x=-4 and x=2.

(*side note: where x+1 is equal to zero, there is a point of removable discontinuity, so (-1, 4/9) is a hole in the graph.)

Where the numerator is equal to zero will give you the x-intercept, so 3 is the x-intercept.

For the y-intercept, substitute zero for x and you will get 3/8.

The answer is A.

Answer 3

f(x) = (x² - 2x - 3) / (x³ + 3x² - 6x - 8)

The numerator is not zero when x = 2, -4, but the denominator is zero at those x-values.

So, x = 2 and x = -4 are both vertical asymptotes.

The numerator and denominator are both equal to zero when x = -1. So, there is a "hole" (removable discontinuity) when x = -1.

f(3) = 0, so the x-intercept is 3, i.e., the point (3,0).

f(0) = 3/8, so the y-intercept is 3/8, I.e., the point (0, 3/8)

(Answer is choice a.)

Note: f(x) = [(x+1)(x-3)] / [(x+1)(x+4)(x-2)]

Answer 4

Solve 𝒙𝟒 + 𝟔𝟒 = 𝟐𝟎𝒙𝟐.

a. 𝑥 = {2, 4}

b. 𝑥 = {−4, −2, 2, 4}

c. 𝑥 = {2𝑖, 4𝑖}

d. 𝑥 = {−4𝑖, −2𝑖, 2𝑖, 4𝑖}