Category Archives: 3.1 Functions

Example: Classify the function as a polynomial function, rational function, or root function, and then find the domain. Write the domain interval notation and set builder notation.

f(x) = root{6}{2-x}

Solution:

Classify the Function

Formal Definition	Practical Way of Identifying
Polynomial Function A polynomial function is a function of the form $f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0$ where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients $a_n , a_{n-1,...,a_1,a_0$ are from the real numbers.	Look for the variables to be in the numerator. (If there is no fraction at all, they are in the numerator.) The variable should not be inside a radical or absolute value. The powers or exponents on the variables should be whole numbers. Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function A rational function is a function of the form where and are polynomial functions and is not equal to zero.	The numerator and denominator are polynomials. Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
Root Function (even index) A root function is a function of the form where n is an even positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an even number. {2, 4, 6, 8, …} The square root is an even index although the index is not written.
Root Function (odd index) A root function is a function of the form where n is an odd positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an odd number. {3, 5, 7, 9, …} The cube root is an odd index.

Since the function f(x) = root{6}{2-x} has a radical and the index is even. This function is a root function.

Find the Domain of a Root Function (Even Index)

Taking the even root of a negative number results in a complex or imaginary number. Since we are interested in real function values, we would like the expression inside the radical to be non-negative ( zero or positive) The root function is defined for any value of the variable where the expression under the radical is non-negative (zero or positive). Find these values by creating an inequality to solve. The inequality is the expression under the radical greater than or equal to zero.

Set the expression under the radical greater than or equal to zero.	0
Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side.	0
Solve by isolating the variable. Start by subtracting 2 on both sides.
Isolate the variable. Continue by dividing both sides by -1. Be sure to reverse the inequality symbol since you are dividing both sides by a negative.

The function values where x<=2 are defined for f(x) = root{6}{2-x} .

In set builder notation, the domain is .

In interval notation, the domain is ( - infty, 2 ]

3.1 Functions, 3.5 Combination of Functions; Composition of Functions

Combination of Function: Evaluation of Division

November 17, 2017 admin

Example:

Evaluate (g/f)(-3) when g(x)=x^2-5 and f(x)=sqrt{x+3}

Solution:

Evaluate the function
Rewrite the expression
Replace x with -3 in each function	$(g/f)(-3)={(-3)^2-5}/{sqrt{-3+3}}$
Simplify	$(g/f)(-3)={(-3)^2-5}/{sqrt{-3+3}}$
Division by zero is undefined	is undefined

(g/f)(-3) is undefined.

3.1 Functions, 3.1 Relations and Functions, 4.6 Rational Functions and Their Graphs, 5.4 Properties of Rational Functions

Finding Domain: Rational Function

July 13, 2017 admin

Example: Classify the function as a polynomial function, rational function, or root function, and then find the domain. Write the domain interval notation and set builder notation.

$h(x) = {x^2+8}/{x^2+x-20}$

Solution:

Classify the Function

Formal Definition	Practical Way of Identifying
Polynomial Function A polynomial function is a function of the form $f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0$ where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients $a_n , a_{n-1,...,a_1,a_0$ are from the real numbers.	Look for the variables to be in the numerator. (If there is no fraction at all, they are in the numerator.) The variable should not be inside a radical or absolute value. The powers or exponents on the variables should be whole numbers. Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function A rational function is a function of the form where and are polynomial functions and is not equal to zero.	The numerator and denominator are polynomials. Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
Root Function (even index) A root function is a function of the form where n is an even positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an even number. {2, 4, 6, 8, …} The square root is an even index although the index is not written.
Root Function (odd index) A root function is a function of the form where n is an odd positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an odd number. {3, 5, 7, 9, …} The cube root is an odd index.

Since the function $h(x) = {x^2+8}/{x^2+x-20}$ has a variable in the denominator and the numerator and denominator are polynomial functions this function is a rational function.

Find the Domain of a Rational Function

Division by zero is undefined. Having a zero as the denominator is equivalent to division by zero thus is also undefined. The rational function is undefined for any value of the variable that gives a zero denominator. Find these values by creating an equation to solve. The equation is the expression in the denominator equal to zero.

Set the denominator equal to zero.
Solve the equation. This equation is a quadratic equation and can be solved by factoring, completing the square or the quadratic formula.
Solve by factoring. Factor the expression on one side.
Use the zero product property and set each factor equal to zero.	or
Solve each equation.	or or or

The values -5 and 4 give a zero value in the denominator, make the function undefined and must be excluded from the domain.

In set builder notation, the domain is delim{lbrace}{ x delim{ |} x!=-5 and x!=4}{rbrace}

In interval notation, the domain is (- infty,-5) union (-5,4) union (4,infty)

3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Quadratic Function

June 26, 2017 admin

Example: Find the difference quotient for f(x)=x^2-9x

The Difference Quotient: {f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
Write the difference quotient for the given function	= ${(x+h)^2-9(x+h)-(x^2-9x)}/h$
Apply the exponent and use the distributive property	= ${(x+h)(x+h)-9x-9h-x^2+9x}/h$
Multiply	= ${x^2+xh+xh+h^2-9x-9h-x^2+9x}/h$
Combine the like terms	= ${x^2+2xh+h^2-9x-9h-x^2+9x}/h$
Combine the like terms. Only terms with h should remain	= ${2xh+h^2-9h}/h$
Divide h into each term	= ${2xh}/h+{h^2}/h-{9h}/h$
Cancel the common h from each term	=

3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

June 6, 2017 admin

Example: Find the difference quotient for f(x)={4x}/{x+5}

The Difference Quotient: {f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
Write the difference quotient for the given function	=
Use the distributive property	=
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator	=
Distribute the common denominator to each fraction in the numerator.	=
Cancel the common factor	=
Multiply the expression in the numerator	= ${4x^2+20x+4xh+20h-4x^2-4xh-20x} /{h(x+5)(x+h+5)}$
Combine like terms	=
Cancel a common h from the numerator and denominator	=