Rational Functions

-Defined as the quotient of two polynomial functions
-Since division by zero is undefined, rational functions have special properties that polynomial functions do not have

Properties of a Rational Function

Asymptotes

Horizontal

Vertical

Intercepts

X-intercept

Y-intercept

Slope

Domain

Range

Intervals

Positive

Negative

Domain

Let the denominator be 0 to find out the domain and the vertical asymptote

x(x)=0

x=0

Therefore the domain is xЄR x 0

Range

The range is yЄR y> 0 since f(x) can never be a zero.

Asymptotes

Horizontal

y=0 is a horizontal asymptote as the function will never touch the x-axis.

Vertical

vertical asymptote is also x=0.

X-intercepts

none

End behaviour

As x approaches positive infinity, f(x) approaches 0 from above. As x approaches negative infinity, f(x) approaches 0 from the above also.
As x approaches 0 from right, f(x) approaches a positive infinity. As x approaches 0 from left, f(x) approaches a positive infinity too.

Slope

The slope is positive and increasing when xЄ(-∞,0) while the slope is negative and increasing when xЄ(0,∞).

Function

The function is positive when xЄ(-∞,0) U (0,∞).
The function is increasing when xЄ(-∞,0) while its decreasing when xЄ(0,∞).

Full analysis

Numeric

Tables

Ordered Pairs

Calculations

Algebraic

Formulas

Solving Equations

Graphical

Verbal

Descriptions

Special Cases

Given an equation, (x-3)(x+2)/ x+2
f(x) = 3
Therefore, f(x) simplifies to a
linear relationship.
This is a special case of a line
that is discontinuous where x cannot be 2.

Factor and reduce where possible

Indicate restrictions on
the variables

Solve Rational Equations Algebraically

1. Factor the expressions in the
numerator and denominator

2. Multiply both sides by the
factored denominator

3. Simply and solve it

Solve Rational Equations using Technology

Graphing Calculator

CAS

Solve Simple Rational Inequalities

Consider Key Features
Of the Graph

Rewrite the right side to 0

Use test points

Determine sign of the expression

Solve Algebraically

Solve a Quadratic Over a Quadratic
Rational Inequality

Interval Table

Number line

Tables

Properties of : Rational Functions of the form f(x) = (ax+b)/(cx+d)

Domain
(2x^2+ 3)/(x+ 3)
Domain : {x є R / x ≠ -3}

Range

x- intercept
: to find x-intercept, y =0
Eg: y=x/(x-2)
x-intercept is 0

y- intercept
: to find y-intercept, x =0
Eg: y=(2x^2+ 4)/(x-2)
y-intercept is -2

Vertical asymptotes
X = restriction of domain

Horizontal asymptotes
If n < m, y="0
If n = m, the horizontal asymptote is y = (coefficient of the x^n term) / (coefficient of the x^m term)
If n > m, there is no horizontal asymptote.