Luokat: Kaikki - asymptotes - domain - range - slope

jonka Wilfred Loh 13 vuotta sitten

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Advanced Function

Rational functions of the form f(x) = (ax+b)/(cx+d) exhibit specific properties, including vertical and horizontal asymptotes, which define their behavior near specific values. The domain of such functions excludes values that make the denominator zero, leading to vertical asymptotes.

Advanced Function

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