Representation Of Rational Functions.pptx

REPRESENTATION OF RATIONAL FUNCTIONS

A rational function can be presented by table of values To construct a table of values for the given rational function, assign values x and substitute to the function then solve. Examples: 1 f(x)= 𝑥 X

-4

-2

f(x)

-0.25 -0.5

-1

-0.5

-0.25 -0.01 0

0.25

0.5 1

2

4

-1

-2

-4

4

2

0.5

0.25

-100

undefined

1

1 f(x)= , 𝑥−2

 Given a function complete the following table of values. x f(x)

-4

-2

-1

0

1

1.5

2

3

4

10

Determine whether the given table of values represents a rational function 1.

2.

3.

x f(x)

2 3 4 -0.25 -0.33 -0.5

x f(x)

-4 3.75

-3 3.67

x f(x)

-4 -7

-3 -5

-2 3

-2 -3

5 6 7 -1 undefined 1 -1 2

-1 -1

8 0.5

9 10 0.33 0.25

0 1 2 undefined 5 4.5

0 1

1 3

2 5

3 4.33

3 7

REPRESENTATION OF RATIONAL FUNCTIONS GRAPHICALLY

A rational function can be presented by graph To graph, construct a table of values for the given rational function, then graph. Examples:

1.

1 f(x)= 𝑥

X

-4

-2

f(x)

-0.25 -0.5

-1

-0.5

-0.25 -0.01 0

0.25

0.5 1

2

4

-1

-2

-4

4

2

0.5

0.25

-100

undefined

1

2.

𝑥−1 f(x)= 𝑥+1

3. f x =

𝑥 2 −3𝑥−10 𝑥

DOMAIN AND RANGE OF A RATIONAL FUNCTIONS

Domain is the set of all acceptable values of x in the function Range is the set of all resulting values of f(x) or y in the function

MATHEMATICAL CONCEPT A rational function is simply a fraction and its denominator must not be equal to 0 for it would not be undefined. Thus, the domain of rational function are real numbers except those that will make the denominator equal to 0.

STEPS IN FINDING DOMAIN OF RATIONAL FUNCTION: Create an equation where denominator is not equal to 0. Solve the equation. Describe the domain.

the

STEPS IN FINDING RANGE OF RATIONAL FUNCTION:

Repalce 𝑓 𝑥 by 𝑦 Solve for 𝑥 −1 Replace 𝑥 by 𝑟 and 𝑦 by 𝑥.

f 𝑥 =

1 𝑥

EXAMPLES

Domain: All real number except zero or written by 𝑥 ∈ ℝ|𝑥 ≠ 0 , ℝ − 0 , ℝ/ 0 Range: All real number except zero f 𝑥 =

1 𝑥−2

Domain: All real number except 2 or written by 𝑥 ∈ ℝ|𝑥 ≠ 2 , ℝ − 2 , ℝ/ 2 Range: All real number except 2

EXAMPLES f 𝑥 =

1 2𝑥+5

Domain: All real number except

𝑥 ∈ ℝ|𝑥 ≠

5 − 2

,

5 − 2

ℝ−

Range: All real number except zero

5 − 2

,

ℝ/

5 − 2

Find the domain and range of the following rational functions:

1.f 𝑥 =

5 𝑥

2 𝑥

2.f 𝑥 = − 3 3. f 𝑥 =

3𝑥−1 𝑥+3

4. f 𝑥 =

4 𝑥−6

5. f 𝑥 =

5𝑥 3𝑥−4

Find the domain and range of the function. Show all your solution/s if necessary.

1.

𝑥−1 f(x)= 𝑥+1

2.

𝑥 f(x)= 𝑥−2

INTERCEPTS OF RATIONAL FUNCTIONS

X-intercept is the abscissa of the point wherein the graph crosses the x-axis Y-intercept is the ordinate of the point wherein the graph crosses the y-axis

MATHEMATICAL CONCEPT

To find x-intercept of the rational function, determine the zeros of the numerator. To find y-intercept of the rational function, determine the value of the function at 𝒙 = 𝟎.

f 𝑥 =

1 𝑥

EXAMPLES

x-intercept: No x-intercept y-intercept: No y-intercept Therefore the graph does not crosses the x and y axes f 𝑥 =

1 𝑥−2

x-intercept: No x-intercept

y-intercept: y-intercept is

1 − 2

Therefore the graph crosses only the y-axis

f 𝑥 =

𝑥 𝑥−2

EXAMPLES

x-intercept: x-intercept is 0 ; y-intercept: y-intercept is 0 Therefore the graph crosses the origin f 𝑥 =

𝑥−4 2𝑥−3

x-intercept: x-intercept is 4

Therefore the graph crosses

4 ; y-intercept: y-intercept is − 3 4 x-axis at 4 and y-axis at − 3

Find the x and y interceptsof the following rational functions:

1.f 𝑥 =

5 𝑥

2 𝑥

2.f 𝑥 = − 3 3. f 𝑥 =

3𝑥−1 𝑥+3

4. f 𝑥 =

4 𝑥−6

5. f 𝑥 =

5𝑥 3𝑥−4

Find the x and y interceptsof the following rational functions:

1.f 𝑥 =

5 𝑥

2 𝑥

2.f 𝑥 = − 3 3. f 𝑥 =

3𝑥−1 𝑥+3

4. f 𝑥 =

4 𝑥−6

5. f 𝑥 =

5𝑥 3𝑥−4

ASYMPTOTES OF RATIONAL FUNCTIONS

An asymptotes of a function is a line wherein the graph of the function gets closer and closer but will not intersect.

The line 𝑥 = 𝑎 is a vertical asymptotes for the graph of the rational function if f(x) either increases of decreases without bound as x approaches from the right or from the left.

MATHEMATICAL CONCEPT

To find vertical asymptote find the real zero of the denominator. To find horizontal asymptote divide 𝑛 both numerator and denominator by 𝑥 , where n is the highest exponent in the given rational function

1. f 𝑥 = 2. f 𝑥 =

3. f 𝑥 = 4. f 𝑥 =

1 𝑥

EXAMPLES Domain: ℝ/ 0 ;Range: ℝ/ 0

1 Domain: ℝ/ 2 ;Range: ℝ/ 2 𝑥−2 𝑥−4 3 1 Domain: ℝ/ ;Range:ℝ/ 2𝑥−3 2 2 𝑥 Domain: ℝ/ 2 ;Range:ℝ/ 1 𝑥−2

Find the horizontal and vertical asymptotes of the following rational functions:

1.f 𝑥 =

5 𝑥 2 𝑥

2.f 𝑥 = − 3 3. f 𝑥 =

3𝑥−1 𝑥+3

Find the horizontal and vertical asymptotes of the following rational functions: 4.

f 𝑥 =

5.

f 𝑥 =

6.

f 𝑥 =

4 𝑥−6 2 𝑥 −5𝑥+6 2𝑥−5 3𝑥 2 −9𝑥+6 𝑥 2 −5𝑥−36

REMEMBER  The horizontal asymptotes of rational function is 𝑦 = 0, if the degree of the polynomial in the numerator is less than the degree of the denominator. The horizontal asymptote of rational function is 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑦= 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 if the degree of the polynomial in the numerator is the same as the degree of the denominator There is no horizontal asymptotes if the degree of the polynomial in the numerator is greater than the degree of the denominator