Category Archives: Sullivan Chapter 3

3.2 Properties of a Function's Graph, 3.3 Properties of Functions

Even, Odd, or Neither

Example: Determine if the function is even, odd, or neither.

f(x)=x^7+x^6

Background Knowledge:

For a function to be even it must fit the following definition.

f(- x)=f(x)

In words, this means that f(- x) must equal the same expression as the original function.

For a function to be odd it must fit the following definition.

f(- x)=-f(x)

In words, this means that f(- x) must equal the opposite expression as the original function.

We will calculate f(- x)  for the given function and determine if it fits either definition.

Solution:

We will calculate f(- x)  forf(x)=x^7+x^6 and determine if it fits either definition.

Calculate f(- x)
 f(- x)=(- x)^7+(- x)^6
Simplify (- x)^7.

  • Group the negatives. Since there are an odd number of negatives multiplied the product is negative.
  • Group the x’s.  Since the bases are the same, 7 x’s multiplied equals x^7
 f(- x)=(- x)^7+(- x)^6
f(- x)=(- x)(- x)(- x)(- x)(- x)(- x)(-x)+(- x)^6
f(- x)=-x^7+(- x)^6
Simplify (- x)^6.

  • Group the negatives. Since there are an even number of negatives multiplied the product is positive.
  • Group the x’s.  Since the bases are the same, 6 x’s multiplied equals x^6

 
f(- x)=-x^7+(- x)^6
f(- x)=-x^7+(- x)(- x)(- x)(- x)(- x)(- x)
f(- x)=-x^7+ x^6

Is the function even? No,  f(- x)=-x^7+ x^6 is not the same as the original function. f(x)=x^7+x^6

Is the function odd? No, f(- x)=-x^7+ x^6 is not the opposite of the original function. -f(x)=-(x^7+x^6)=-x^7-x^6

This function is not even or odd so we categorize it as “neither.”

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3.1 Functions, 3.1 Relations and Functions

Finding Domain: Radical/Root Function (Even Root)

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

f(x)={p(x)}/{q(x)}

where p(x) and q(x) are polynomial functions and q(x) 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.
2-x 0
Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side. 2-x 0
Solve by isolating the variable.  Start by subtracting 2 on both sides. 2-x 0
2-2-x 0-2
0-x -2
-x -2
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. -x -2
{-x}/-1 {-2}/-1
x<=2

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]

3.4 Transformation of Functions, 3.5 Graphing Techiques: Transformations

Graphing by Transformations: Quadratic

Example: For the function below.  Graph using transformations.

f(x)=-(x-3)^2+4

First we must examine the base function y=x^2

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.  Substitute the standard set of x-values into the base function to get the base graph.

xy
-2(-2)^2=4
-1(-1)^2=1
0(0)^2=0
1(1)^2=1
2(2)^2=4

The graph below shows the points plotted and the line that connects them.  The domain is (- infty, infty)  and the range is [0, infty)

 

Analyze the transformations.

f(x)=-(x-3)^2+4

The -3 inside the square shifts the graph right 3 units.

The – in the front of the base reflects the graph over the x-axis.

 

The +4 outside the square shifts the graph up 4 units.

You can see the graph after the transformations.

The domain is (- infty, infty)  and the range is (- infty, 4]

Here is a video example of a transformation of a square function.