MATH B1A/L – PRE-CALCULUS I
Unit 1 – Objective 1 Notes
1.1 – Functions
A function is a mapping from one set of items, A, to another set of items, B, such
that each element in set A maps to exactly one element (a unique element) in set
B. In other words, for each input value for the function, there is a unique output
associated with the input value.
{Set A is known as the domain of the function, and set B is known as the range of the function.}
EXAMPLE 1 For each of the following, is the mapping a function?
(a)
Set A
Set B
Set A
Set B
Set A
Set B
(b)
(c)
Other names for domain variables: input, x, control variable, independent variable
Other names for range variables: output, y, dependent variable
Vertical Line Test
A test to check whether a graph is that of a function.
If any vertical line crosses the graph more than _______, then the graph __________ a
function.
=>
EXAMPLE 2 Tell whether the graphs are functions or not.
(a)
(b)
(c)
Intervals of Increasing and Decreasing
Stating the domain, range, and intervals of increasing and decreasing (and in
some cases, constant) is given in interval notation. However, only the range is
given as an interval of y-values (from bottom to top). All of the others are given
as an interval of x-values (from left to right).
EXAMPLE 3 State the domain, the range, and the intervals of increasing and decreasing.
(a) The graph from part (a) above.
(b) The graph from part (b) above.
i. Domain:
i. Domain
ii. Range
ii. Range
iii. Increasing
iii. Increasing
iv. Decreasing
iv. Decreasing
Evaluating Functions
Plug in a number or expression for x and evaluate (simplify) to get the output, y.
EXAMPLE 4 Evaluate the following functions at the given value or expression.
(a) If f ( x) = 3 x + 2 , find:
(b) If g ( x) = x 2 − 3x − 10 , find:
i. f (5)
i. g (7)
ii. f ( −4)
ii. g ( −2)
iii. f (2 x − 3)
iii. g ( x + 4)
 x 2 + 1, if x  1
(c) If h( x) = 
, find:
| 5 − 2 x |, if x  1
i. h (0)
ii. h(−3)
iii. h (7)
iv. h (1)
Classifying a function as Even, Odd, or Neither
A function f is even, if for every x in the domain of f, f (− x) = f ( x) .
Even functions are also symmetric about the y-axis.
A function f is odd, if for every x in the domain of f, f (− x) = − f ( x) .
Odd functions are also symmetric about the origin.
EXAMPLE 5 Classify each of the following functions as even, odd, or neither. Show work!
(a) f ( x) = x 2 − 5
(b) g ( x) = x 2 − 3x + 1
(c) h( x) = 3 − x 2
(d) f ( x) =
7
x −x
3