Rational functions

Question

"x²-4"/"x²-4x+3"

Given this fraction, find the

A. Domain

B. Vertical asymptotes

C.Horizontal asymptote

D. X and Y intercepts

E. the range

 

Please walk me through how to find these things.

Thanks!

Answer 1

A)

 

The domain is the set of x values in which a function is defined.  Set the denominator equal to zero.  This will give use the domain as well as vertical asymptotes.

 

x² - 4x + 3 = 0

 

Solve for x from this equation.

 

(x - 1)(x - 3) = 0

 

x = 1       and        x = 3

 

 

Domain is   (-∞, 1)∪(1, 3)∪(3, ∞)

 

 

B)

 

Use the answer from the previous part.

 

x = 1       and       x = 3

 

 

C)

 

We examine the degree of the numerator and the degree of the denominator.  Since degree of the numerator and the degree of the denominator are the same, we only divide coefficients of the leading term of numerator and denominator.

 

Therefore, the horizontal asymptote is         y=1

 

 

D)

 

To find the x-intercepts, set the numerator equal to zero and solve for x.

 

To find the y-intercept, set x=0 and evaluate the expression.

 

 

E)

 

We use the answer from part C.  The range is the set of y values in which a function is defined.

 

Range is  (-∞, 1)∪(1, ∞)