Advanced Function

Mais informações

research methods

por amany elghamry

math 1

por j k mack

Attitude

por Muhammad Hasnan Mohd Salim

Postive Feedback

por Erica Cave

Rational Functions

3.3 Rational Functions of the Form f(x) (ax+b)/(cx+d)

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

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.

Vertical asymptotes X = restriction of domain

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

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

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

3.4 Solve Rational Equations and Inequalities

Solve a Quadratic Over a Quadratic Rational Inequality

Interval Table

Number line

Solve Simple Rational Inequalities

Solve Algebraically

Consider Key Features Of the Graph

Determine sign of the expression

Use test points

Rewrite the right side to 0

Solve Rational Equations using Technology

CAS

Graphing Calculator

Solve Rational Equations Algebraically

3. Simply and solve it

1. Factor the expressions in the numerator and denominator

2. Multiply both sides by the factored denominator

3.5 Making Connections With Rational Functions and Equations

Special Cases

Factor and reduce where possible

Indicate restrictions on the variables

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.

Full analysis

Verbal

Descriptions

Graphical

Algebraic

Solving Equations

Formulas

Numeric

Calculations

Ordered Pairs

Tables

3.2 Reciprocal of a Quadratic Function Example : y= 1/x^2

Function

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

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

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.

X-intercepts

none

Asymptotes

Vertical

vertical asymptote is also x=0.

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

Range

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

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

3.1 Reciprocal of a Linear Function

Properties of a Rational Function

Intervals

Negative

Positive

Range

Domain

Slope

Intercepts

Y-intercept

X-intercept

Asymptotes

Vertical

Horizontal

-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