A function 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 called the domain of the function.
The set B is called the range of the function.
X
Y
2
A
4
B
1
5
2
10
8
3
15
1
4
10
1
y – 3x = 10
3
6
7
Example: Determine whether the relation represents y as
a function of x.
a)
{(-2, 3), (0, 0), (2, 3), (4, -1)} Function
b)
{(-1, 1), (-1, -1), (0, 3), (2, 4)} Not a Function
y depends on value of x,
•
•
•
•
y is dependent variable
x is the independent variable
f(x) is dependent variable
y and f(x) are the same
Tests for function
• vertical line test
– no vertical line intersects graph at
more than one point
• If there is a y variable with largest
exponents being odd, usually it is a
function (even exponents usually are
not)
• All first members of ordered pairs are
different
Is y a function of x?
• is 3x + y = 5? Solve for y
• y = 5 - 3x
(yes, y is a function of x)
y is not a function of x,
• y2 = 25 - x2
• if x = 3 then y = 4 or y = -4
y is not a function of x (even
exponent)
y   25  x
2
f(x) notation
•
•
•
•
•
•
f(x) = 3x - 1
used to name a function
f is name of function
x is independent variable
3x - 1 is rule used to evaluate
y = f(x)
To determine if equation defines a function,
• 1. solve equation for dependent variable (y)
• 2. determine if each single value of independent
variable (x) produces exactly one value of dependent
(y) variable
– * normally x is independent variable while y is
dependent
To evaluate a function f (x) at x = a, substitute the
specified value a for x into the given function.
2
Example: Let f (x) = x – 3x – 1. Find f (–2).
2
f (–2) = (–2) – 3(–2) – 1
f (–2) = 9
2
Example: Let f (x) = 4x – x . Find f (x + 2).
f (x + 2) = 4(x + 2) – (x + 2)2
f (x + 2) = 4x + 8 – (x2 + 4x + 4)
2
f (x + 2) = 4x + 8 – x – 4x – 4
f (x + 2) = 4 – x2
In a piecewise-defined function, you are given two or more functions
to work with followed by defined domains for each function. You
need to decide which domain the value you are using fall in and use
your value with that function.
Ex. Evaluate the function when x = 5 and -3
x 2  1, x  0
f ( x)  
x  1, x  0
When x = 5, we us the bottom function 5 – 1 = 4 so f(5) = 4
When x = –3, we us the top function
(-3)2 + 1 = 10 so f(-3) = 10
The domain of a function f is the set of all real
numbers for which the function makes sense.
Example: Find the domain of the function
f (x) = 3x +5
Domain: All real numbers
Example: Find the domain of the function f ( x)  x  3
The function is defined only for x-values for which x – 3  0.
Solving the inequality yields
x–30
x3
Domain: {x| x  3}
Example: Find the domain of the function
x2
g ( x)  2
x 1
The x values for which the function is undefined
are excluded from the domain. The function is
2
undefined when x – 1 = 0.
2
x –1=0
(x + 1)(x – 1) = 0
x=1
Domain: {x| x   1}
Restrictions on domain
Domain usually all real numbers
f(x) = (3 - x)
3-x > 0 so x < 3
written {x|x < 3} = D
Another exclusion of numbers
g(x) = 4/ (x-2),
x2
so {x|x  2}=D