Rational Functions

)(

)()(

xq

xpxf A function of the form

where p(x) and q(x) are polynomial functions and q(x) ≠ 0.

5x

x)x(f

4x3x

4x)x(g

2

Examples:

(MCC9-12.F.IF.7d)

Graphs of Rational Functions

may have breaks in Continuity.

Breaks in Continuity can appear as:

1. Vertical Asympotes

2. Point Discontinuity

(A hole in the graph)

(MCC9-12.F.IF.7d)

Vertical AsymptoteIf a rational expression is written in simplest form and the function is undefined for x = a, then x = a is a vertical asymptote.

2x

x)x(f

Example:6

4

2

-2

-4

-5 5x = - 2is vertical

asymptote.

(Note: Set the denominator equal to zero

& solve for x.)

Point DiscontinuityIf the original function is undefined for x = a but the rational expression of the function in simplest form is defined for x = a, then there is a hole in the graph at x = a.

2x

)1x)(2x()x(f

Example:4

2

-2

-4

-6

-5 5

Point of Discontinuityas x = -2

(Note: If a factor cancels in the top & bottom, set it equal to zero & solve for

x.)

Finding Horizontal AsymptotesFinding Horizontal Asymptotes

for for RationalRational Functions Functions

Given a rational function: f (x) = p(x) am xm + lower degree terms

q(x) bn xn + lower degree terms=

Let am be the leading coefficient of the numerator and m be the degree of the

numerator.

Let bn be the leading coefficient of the denominator and n be the degree of the

denominator.(MCC9-12.F.IF.7d)

• If m > n, then there are no horizontal asymptotes.

• If m < n, then y = 0 is a horizontal asymptote.

• If m = n, then y = am is a horizontal asymptote. bn

(MCC9-12.F.IF.7d)