Functions and Function Notation

1
Functions and Function Notation

• Section 3.1

2
Objectives

• Given an equation determine if y is a function
  of x.
• Determine if a graph represents a function.
• Find the domain and range of a function.
• Find the value of a function
• Evaluate the difference quotient
• Graph functions.

3
Inputs and OutputsConsider y x2 1

• If x 2, then y 5.
• x is the input, y is the output.

4
Function- Any rule that assigns one value y to
each number x

• Image of x - is the value of y (output) that
  corresponds to x (input)
• The value of y(ouput) depends on the number
  x(input), therefore, y is the dependent variable
  and x is the independent variable.
• The values x can be is the domain of the
  function.
• The values y will be is the range of the
  function.
• (The images of x.)

5
Visualizing functions by mapping.
X - Domain
Y-Range

• Function
• Function
• Not a Function

1 2 3
1 2 3
1 2 3
3
1 2
3
6
To Determine if an Equation Represents a Function

• Solve each equation for y in terms of x
• If more than one value of y is the result of any
  one value of x , then the equation is not a
  function.

7
Determine in which equations y is a function of
x.

• y x
• y 3x 1
• y x 1
• y - 6 6

• x - 2 y
• x y
• y2 4x -1
• x y 6

8
Vertical line test - If any vertical line
intersects the graph exactly once then the graph
represents a function
9
Does the graph represent a function?
10
Domain and Range

• Domain- The set of all the x-coordinates of the
  ordered pairs.
• Range The set of all the y-coordinates of the
  ordered pairs.

11
Graph of a Function - in the xy-plane is the set
of all points (x,y) where x is in the domain of
f and y is in the range of f.
12
Finding the Domain of a Function

• Domain is the set of all real numbers for which
  the value of f(x) is a real number
• Exclude all real numbers that cause division by
  zero.
• Exclude all real numbers that result in an even
  root of a negative number.

13
Finding the range of a function

• Range is set of y-values.
• One way to find the restrictions on y is to solve
  the equation for x. Then
• Exclude all real numbers that cause division by
  zero.
• Exclude all real numbers that result in an even
  root of a negative number.

14
Find the domain and range of each function.
15
Evaluating functional notation.Y f(x) is read
as y is a function of x

• Let f(x) 10 - 3x
• f(2) means to evaluate 10 - 3x when x 2
• f(2) 10 - 3(2)
• 10 - 6
• f(2) 4
• The image of 2 is 4. This represents the ordered
  pair (2,4)

16
Let g(x) 3x2 2x - 7

• Find g( 4)
• Find g(-y2)

• Find g( a 2)

17
Difference Quotient

• Evaluate the difference quotient if
  f(x) 6x - 3

18
Graph the function
19
Graph the function
20
Graph the function
21
Linear Function is a function determined by an
equation of the form f(x) mx b

• The velocity of a falling object is a linear
  function of the time t it has been falling.
• If v 15 when t 0 and v 79 when t 2
  express v as a function of t.

22
Not all equations define functions!

• Graph y2 x