Functions

1
Functions

• Section 2.1

2
Functions

• Weve been studying relationships between two
  variables x and y. Now we turn our attention to
  special relationships that well call functions.
• y -2x 3 is a function because for every one
  x value (input), there is only one y value
  (output) associated with it.
• is not a function. Put a value in for x and
  youll get two outputs for y.

3

• Functions are unambiguous. Everyone who puts in
  an x value will get out the same y value as
  everybody else
• Think about the relationship y -2x 3
  everybody who puts in 5 for x, gets out -7 for y.
  
• We can discuss the relationship easier because
  its a function and we all get the same thing for
  y given an x value.
• This is not the case with . Put 5 in for
  x, and well get two different y values out, -3
  and 3.
• Lets look at functions graphically.

4

• Look at the graph of
• y -2x 3.
• For every x value on the graph, there is only one
  y value associated with it. So the relationship y
  -2x 3 is a function.

5
Look at the graph of . For nearly
every x value on the graph, there is more than
one y value associated with it. So the
relationship is not a function.
This x value has more than one y value.
6
Vertical line test

• If you draw a vertical line through a graph
  anywhere and it never hits more than one point,
  then the relationship is a function.

Any vertical line (representing a single x value)
will only hit one spot on the graph (representing
a single y value) if the relationship is a
function.
7
Counterexamples non-functions
These are not functions. Notice for the x value
highlighted by the vertical line, there is more
than one y value. In the case of x 2 y 2 25
in the middle, when x is 3, y could be 4 or -4.
8
Functional notation

• Is y 2x 2 3x 5
• a function?
• Look at its graph. Does it pass the vertical line
  test?

9

• Since y 2x 2 3x 5 is a function, we will
  write f (x) 2x 2 3x 5 to indicate that y is
  a function of x. So we write y f (x) or
• y is a function of x

We can think of this notation as a rule. It tells
you what to do to the x value (or any value in
the parentheses) to get the y value. f (x) 2x
2 3x 5 means we square it, multiply that by
2, add 3 times it, then subtract 5. So lets do
that!
10
expl Let f (x) 2x 2 3x 5. Find f (2), f
(-4), and f (x - 1).
You apply the rule to the number x. Whatever the
number is, you square it, multiply by 2, add 3
times it, then subtract 5. Mind your order of
operations.
11

• You can think of functions in a few different
  ways.
• 1. a relationship between two variables, x and y
• 2. a rule that tells you what to do to an x
  value to get a y value
• 3. a machine that produces a y value when you
  input an x value

12
Domain / Range
domain of function the set of x values that make
sense or work in the equation (that produce a y
value) Notice when x - 4, . But
that does not exist in the real numbers. Since
this is not an acceptable value for y, we say -4
is not in the domain. The domain consists of only
those x values that work in the equation. This -4
does not work.

• Expl Consider

13

• We usually denote the domain of a function as
  the set of numbers that work for x in the
  equation. Usually looking at the graph will help.

Notice the only x values that correspond with
points on the graph are 3 and greater. So that
means only these x values work in the equation.
So the domain of this function is .
Also, notice the graph does not include x -4,
as we saw -4 was not in the domain earlier.
14

• You can also think through domain algebraically
• We start with . Since we cannot take the
  square root of negative numbers, we know that
  . We solve this to get or or .

15
Now, there are two things to worry about when
finding domain square roots of negative numbers
and division by zero. If an x value would cause
either of these two things to happen, then we
exclude this x value from the domain. In our
example of , when x -4, . But,
you cannot take the square root of -7 in the real
numbers, so we say -4 cannot be in the domain.
16

• Range the set of y values that you can get as
  output values

Notice how only positive and zero y values are
represented by the graph. And since y is the
square root of some number, y cannot be negative,
so we say the range of this relationship is
or or .
17

• expl Find domain of .
• -- No square roots no worries there
• -- If x 1, the denominator, were zero, then
  wed be dividing by zero. So solve x 1 0 to
  get x 1. Well exclude this x value from the
  domain.
• Domain all real numbers except 1 or .

Remember there are two things to worry about when
finding domain 1. square roots of negative
numbers, and 2. division by zero.
18

• expl Determine whether the equation is a
  function.
• a. b.

Think about putting 4 in for x, youd get just
one y. Look at its graph too. Its a function!
Solve the equation for y. Youll get Not a
function!
19
Linear function Since y mx b, the straight
line you may have seen earlier, is a function, we
will use the notation f(x) mx b.
2.1 homework 3, 5, 15, 23, 25, 31, 33, 35, 37,
41, 43, 47, 49, 50, 51, 53, 61, 68, 73, 79, 82,
85, 93