Answer:
* Domain all real value of x except x = 4
* Vertical asymptotic x = 4
* Roots are -4 and 3
* Y-intercept is 3
* No horizontal asymptotic
* There is no holes
* O.A is y = x + 5 doesn't cross the function
Explanation:
* ∵ f(x) =( x² + x - 12)/x - 4
∵ x - 4 = 0 ⇒ x = 4
∴ The domain of the function is all real number except x = 4
* The vertical asymptotic is when denominator = 0
∴ x - 4 = 0 ⇒ x = 4
∴ The vertical asymptotic is x = 4
* The roots of the function is the value of x when y = 0
∴ x² + x -12 = 0 ⇒ (x + 4)(x - 3) = 0
∴ x = -4 and x = 3
∴ The roots are -4 and 3
* Y-intercept means x = 0
∴ f(0) = 0 + 0 - 12/0 - 4 = -12/-4 = 3
∴ The y-intercept = 3
* ∵ The degree of numerator is greater than the degree of denominator
∴ There is no horizontal asymptotic
* ∵ The hole is the value of x which makes the denominator and the
numerator = 0
∴ There is no holes in this function
* to find O.A ⇒ x² + x -12 ÷ x - 4 = x + (5x - 12)/x - 4
= x + 5 + 8/x - 4
∴ The O.A is y = x + 5
To check if it will cross the function ⇒ (x - 4)(x + 5) = x² + x - 20
Compare it with x² + x - 12 we will find -20 ≠ -12
∴ It will not cross the function