Rational Function: Functions and Their Modeling - Mathematics (Grade 11)

What is a Rational Function?

Have you ever seen fractions in mathematics? Well, rational functions are similar to fractions, but more interesting because they involve variables!

A rational function is a function in the form of a fraction, where both the numerator and denominator are polynomial functions. Simply put, a rational function can be written as:

f(x) = \frac{P(x)}{Q(x)}

Where:

$P(x)$ is a polynomial in the numerator
$Q(x)$ is a polynomial in the denominator
$Q(x) \neq 0$ (denominator cannot be zero)

Examples of Rational Functions in Life

Let's look at a real example to better understand rational functions.

Chicken Coop Problem:

Mr. Budi wants to build a rectangular chicken coop with an area of 100 m². He wants to know the relationship between the length and width of the coop.

If the length of the coop is $x$ meters, then:

Area = length × width = 100
$x \times \text{width} = 100$
Width = $\frac{100}{x}$

The function $f(x) = \frac{100}{x}$ is an example of a rational function!

Types of Rational Functions

Simple Rational Function

The simplest form of a rational function:

f(x) = \frac{k}{x}

Where $k$ is a constant. Example: $f(x) = \frac{5}{x}$

Linear Rational Function

Both numerator and denominator are linear functions:

f(x) = \frac{ax + b}{cx + d}

Example: $f(x) = \frac{2x + 3}{x - 1}$

Quadratic Rational Function

Involves quadratic functions in the numerator or denominator:

f(x) = \frac{x^2 + 2x + 1}{x - 3}

Domain of Rational Functions

The domain of a rational function is all values of $x$ that make the function defined. Remember, the denominator cannot be zero!

How to find the domain:

Find values of $x$ that make the denominator = 0
The domain is all real numbers except those values

Example: Determine the domain of $f(x) = \frac{x + 2}{x - 3}$

Solution:

Denominator is zero when: $x - 3 = 0$
So: $x = 3$
Domain: $D_f = \{x | x \neq 3, x \in \mathbb{R}\}$

Simplifying Rational Functions

Rational functions can be simplified by finding common factors in the numerator and denominator.

Without Factoring

Simplify: $f(x) = \frac{6x^2}{3x}$

Solution:

f(x) = \frac{6x^2}{3x}

= \frac{6x \cdot x}{3x}

= \frac{6x}{3} \cdot \frac{x}{x}

= 2x

With Factoring

Simplify: $f(x) = \frac{x^2 - 4}{x - 2}$

Solution:

f(x) = \frac{x^2 - 4}{x - 2}

= \frac{(x + 2)(x - 2)}{x - 2}

= x + 2

Note: $x \neq 2$ (from the original domain)

Operations on Rational Functions

Addition and Subtraction

Just like regular fractions, we need to find a common denominator first!

Example: $\frac{2}{x} + \frac{3}{x + 1}$

Solution:

\frac{2}{x} + \frac{3}{x + 1}

= \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)}

= \frac{2(x + 1) + 3x}{x(x + 1)}

= \frac{2x + 2 + 3x}{x(x + 1)}

= \frac{5x + 2}{x(x + 1)}

Multiplication

Multiply numerator with numerator, denominator with denominator:

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Example: $\frac{x + 1}{x} \times \frac{2x}{x - 1}$

Solution:

\frac{x + 1}{x} \times \frac{2x}{x - 1} = \frac{2x(x + 1)}{x(x - 1)} = \frac{2(x + 1)}{x - 1}

Division

Remember, dividing = multiplying by the reciprocal:

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Exercises

Determine the domain of $f(x) = \frac{x + 3}{x^2 - 9}$
Simplify $f(x) = \frac{x^2 - 1}{x + 1}$
Calculate $\frac{1}{x - 1} - \frac{2}{x + 1}$
A car travels 300 km. If the average speed is $v$ km/h, write the travel time function in terms of $v$ .

Answer Key

Answer 1:

x^2 - 9 = 0

(x + 3)(x - 3) = 0

x = -3 \text{ or } x = 3

Domain: $D_f = \{x | x \neq -3, x \neq 3, x \in \mathbb{R}\}$

Answer 2:

f(x) = \frac{x^2 - 1}{x + 1}

= \frac{(x + 1)(x - 1)}{x + 1}

= x - 1

With the condition $x \neq -1$

Answer 3:

\frac{1}{x - 1} - \frac{2}{x + 1}

= \frac{(x + 1) - 2(x - 1)}{(x - 1)(x + 1)}

= \frac{x + 1 - 2x + 2}{(x - 1)(x + 1)}

= \frac{-x + 3}{(x - 1)(x + 1)}

Answer 4:

Time = Distance ÷ Speed

t(v) = \frac{300}{v}

What is a Rational Function?

Have you ever seen fractions in mathematics? Well, rational functions are similar to fractions, but more interesting because they involve variables!

A rational function is a function in the form of a fraction, where both the numerator and denominator are polynomial functions. Simply put, a rational function can be written as:

f(x) = \frac{P(x)}{Q(x)}

Where:

$P(x)$ is a polynomial in the numerator
$Q(x)$ is a polynomial in the denominator
$Q(x) \neq 0$ (denominator cannot be zero)

Examples of Rational Functions in Life

Let's look at a real example to better understand rational functions.

Chicken Coop Problem:

Mr. Budi wants to build a rectangular chicken coop with an area of 100 m². He wants to know the relationship between the length and width of the coop.

If the length of the coop is $x$ meters, then:

Area = length × width = 100
$x \times \text{width} = 100$
Width = $\frac{100}{x}$

The function $f(x) = \frac{100}{x}$ is an example of a rational function!

Types of Rational Functions

Simple Rational Function

The simplest form of a rational function:

f(x) = \frac{k}{x}

Where $k$ is a constant. Example: $f(x) = \frac{5}{x}$

Linear Rational Function

Both numerator and denominator are linear functions:

f(x) = \frac{ax + b}{cx + d}

Example: $f(x) = \frac{2x + 3}{x - 1}$

Quadratic Rational Function

Involves quadratic functions in the numerator or denominator:

f(x) = \frac{x^2 + 2x + 1}{x - 3}

Domain of Rational Functions

The domain of a rational function is all values of $x$ that make the function defined. Remember, the denominator cannot be zero!

How to find the domain:

Find values of $x$ that make the denominator = 0
The domain is all real numbers except those values

Example: Determine the domain of $f(x) = \frac{x + 2}{x - 3}$

Solution:

Denominator is zero when: $x - 3 = 0$
So: $x = 3$
Domain: $D_f = \{x | x \neq 3, x \in \mathbb{R}\}$

Simplifying Rational Functions

Rational functions can be simplified by finding common factors in the numerator and denominator.

Without Factoring

Simplify: $f(x) = \frac{6x^2}{3x}$

Solution:

f(x) = \frac{6x^2}{3x}

= \frac{6x \cdot x}{3x}

= \frac{6x}{3} \cdot \frac{x}{x}

= 2x

With Factoring

Simplify: $f(x) = \frac{x^2 - 4}{x - 2}$

Solution:

f(x) = \frac{x^2 - 4}{x - 2}

= \frac{(x + 2)(x - 2)}{x - 2}

= x + 2

Note: $x \neq 2$ (from the original domain)

Operations on Rational Functions

Addition and Subtraction

Just like regular fractions, we need to find a common denominator first!

Example: $\frac{2}{x} + \frac{3}{x + 1}$

Solution:

\frac{2}{x} + \frac{3}{x + 1}

= \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)}

= \frac{2(x + 1) + 3x}{x(x + 1)}

= \frac{2x + 2 + 3x}{x(x + 1)}

= \frac{5x + 2}{x(x + 1)}

Multiplication

Multiply numerator with numerator, denominator with denominator:

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Example: $\frac{x + 1}{x} \times \frac{2x}{x - 1}$

Solution:

\frac{x + 1}{x} \times \frac{2x}{x - 1} = \frac{2x(x + 1)}{x(x - 1)} = \frac{2(x + 1)}{x - 1}

Division

Remember, dividing = multiplying by the reciprocal:

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Exercises

Determine the domain of $f(x) = \frac{x + 3}{x^2 - 9}$
Simplify $f(x) = \frac{x^2 - 1}{x + 1}$
Calculate $\frac{1}{x - 1} - \frac{2}{x + 1}$
A car travels 300 km. If the average speed is $v$ km/h, write the travel time function in terms of $v$ .

Answer Key

Answer 1:

x^2 - 9 = 0

(x + 3)(x - 3) = 0

x = -3 \text{ or } x = 3

Domain: $D_f = \{x | x \neq -3, x \neq 3, x \in \mathbb{R}\}$

Answer 2:

f(x) = \frac{x^2 - 1}{x + 1}

= \frac{(x + 1)(x - 1)}{x + 1}

= x - 1

With the condition $x \neq -1$

Answer 3:

\frac{1}{x - 1} - \frac{2}{x + 1}

= \frac{(x + 1) - 2(x - 1)}{(x - 1)(x + 1)}

= \frac{x + 1 - 2x + 2}{(x - 1)(x + 1)}

= \frac{-x + 3}{(x - 1)(x + 1)}

Answer 4:

Time = Distance ÷ Speed

t(v) = \frac{300}{v}

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