Question

Enter the equations of the asymptotes for the function

Answer 1

x = 7 and y = 2

Explanation:

the denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote

solve x - 7 = 0 ⇒ x = 7 is the asymptote

horizontal asymptotes occur as

[tex]lim( x → ± ∞), f(x) → c ( where c is a constant )

divide terms on numerator/ denominator by x

f(x) = (3/x/x/x -7/x ) + 2

as x → ± ∞, f(x) → 0 / 1 - 0 + 2 = 2

y = 2 is the asymptote

Answer 2

Answer: Vertical asymptote is x = 7

Horizontal asymptote is y = 2

Explanation:

The vertical asymptote is the restriction on the domain (x-value). Since the denominator cannot be zero ⇒ x - 7 ≠ 0 ⇒ x ≠ 7 so the vertical asymptote is at x = 7.

The horizontal asymptote (H.A.) is the restriction on the range (y-value). There are three rules that determine the horizontal value which compare the degree of the numerator (n) with the degree of the denominator (m):

• If n > m , then there is no H.A. (use long division to find the slant asymptote)
• If n = m , then the H.A. is the coefficient of n ÷ coefficient of m
• If n < m, then the H.A. is y = 0

In the given problem, n = 0 and m = 1 ⇒ n < m ⇒ H.A. is y = 0

Since there is a vertical shift of +2 units, the H.A. is y = 0 + 2 ⇒ y = 2