Combinations of Functions

Combinations of Functions:

All of these are Arithmetic combinations of functions

Let f(x) =2x+3
let g(x) =

Creating new functions - examples

(f+g)(x) =f(x)+g(x)

=2x+3+

(f-g)(x) =f(x)-g(x)

= (2x+3)-( )

= -+2x+3

(fg)(x) =f(x)*g(x)

= (2x+3)( )

= 2+3

(f/g)(x)= ------> don’t forget to find the domain after dividing!

Remember-denominator cannot be zero!

= (2x+3)/( )

1.2

Finding Domain and Range of a Function

• Algebraic Solution

Note: Because the expression is under a radical it must end up being ≥ 0 otherwise it would be imaginary.

Step 1. Set the equation as  ≥ 0

             ≥ 0

Step 2. Square both sides to get rid of the radical. It should end up being 

            x-4 ≥ 0

Step 3. Isolate the variable

          x-4+4≥ 0+4

                  x ≥ 4

x is all real numbers greater than or equal to zero [4,∞)

• Graphical Solution

f(x)=

The graph for this function looks like this

To find the domain and range simply look at the real line

As you an see the range is all real numbers greater than or equal to zero & the range is all real numbers greater than or equal to 4

Even and Odd Functions

Function 1

Find if f(x) = –3x² + 4 even, odd, or neither.

Plug in f(-x) and solve

f(–x) = –3(–x)² + 4 

      = -3(-x)(-x) +4

         = –3(x²) + 4          = –3x² + 4

The function is the same as it started out as therefore it's an even function, the graph would end up looking like this with the y axis as the line of symmetry

Function 2

Is f(x) = 2x³ – 4x even, odd, or neither?

Same as last time, plug in f(–x)

f(–x) = 2(–x)³ – 4(–x) 

         = 2(–x³) + 4x          = –2x³ + 4x

As you can see the result is the exact opposite of the original, which means that the function is odd. When graphed it would look like this with the point of origin as the line of symmetry 

Function 3

Is f(x) = 2x³ – 3x² – 4x + 4 odd, even or neither?

Plug in f(–x)

f(–x) = 2(–x)³ – 3(–x)² – 4(–x) + 4          = 2(–x³) – 3(x²)  + 4x + 4          = –2x³ –3x² + 4x + 4

It's not the same or opposite of the original equation which means it is neither and a graph for it would not follow the symmetry on the y axis or the point of origin.

Domain and Range of Functions, Difference Quotient

Domain = all possible x-values
Range = all possible y-values

Domain

When finding the domain, check for:

• negative square roots


• denominators that equal zero
  

Examples

1.) In equations with fractions, the domain is determined only by the denominator. It is important to make sure that it will not equal zero.

Start by setting the equation equal to zero and solving.

These are the values that will make the denominator equal zero, so the domain of the function is all numbers except 2 and -1.

2.) In equations with a square root, it is important to make sure that we are not finding the square root of a negative number.

To start, set the equation as greater than or equal to zero and solve.

This is the lowest x-value that can be used before the equation becomes negative. Therefore, the domain of the equation is all values equal to or less than three halves.

Range

To find the range, graph the equation to see all the different y-values.

In this example, we can see that the lowest y-value on the line is -1, and that the lines extend upwards indefinitely.

Therefore, the range is

Difference Quotient

http://www.youtube.com/watch?v=1O5NEI8UuHM

Ch1- Functions

Function- f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the doman ( or set of inputs) of the function f, and the set B contains the range ( or set of outputs).

Functions can be expressed Verbally, Numerically, Graphically, and Algebraically
Use the vertical line test to check if there is a function on a graph. If the line passes 2 points then there are two of the same x values, therefore it is not a function.


Function Notation-
y is a function of x
y = f (x) ↔ The point (x, y) is on the graph f

Example- f (x)= 3x+10
f (2)= 3(2)+10
f (2)=16