Help me find these for this: y=5x+3/x-2

Question

1.) Domain 2.) Vertical/Horizontal Intercepts 3.) Vertical/ Horizontal Asymptotes 4.) Hole(s)?

Answer 1

The rational function y = (5x + 3)/(x - 2)

1) Domain

We know that all possible values of x is domain of a function.

A rational function is simply fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

x - 2 ≠ 0

x ≠ 2

So the domain of the function all real numbers except 2.

Domain set is {x∈R:x ≠ 2}.

Answer 2

2) y = (5x + 3)/(x - 2)

The points where the function crosses the vertical and horizontal axes are known as the vertical and horizontal intercepts of the function.

To find horizontal intercept (x intercept) substitute y = 0 in the function.

(5x + 3)/(x - 2) = 0

5x + 3 = 0

x = - 3/5

To find vertical intercept(y intercept) substitute x = 0 in the function.

y = [5(0) + 3]/[(0) - 2]

y = - 3/2.

Answer 3

3) y = (5x + 3)/(x - 2)

Vertical asymptote can be found by making denominator = 0.

x - 2 = 0

x = 2

Vertical asymptote is at x = 2.

To find horizontal asymptote, first find the degree of the numerator and  the degree of denominator.

Degree of the numerator = 1 and the degree of denominator = 1.

Since the degree of the numerator is equal to the degree of the denominator,horizontal asymptote is the ratio of the leading coefficient of numerator and denominator.

Leading coefficient of numerator = 5, leading coefficient of denominator = 1

y = 5 is the horizontal asymptote.

Answer 4

4) The rationa function y = (5x + 3)/(x - 2)

There is no common factors to cancel in the numarator and denominator of the above rational function.

Holes : none.