Pre-Calculus
Unit 1
Section 1.1 - Functions
Real numbers are used to describe quantities such as money and distance. The set of real numbers ℛ
subsets of numbers:
Notation
Interval notation uses inequalities to describe subsets of real numbers.
• [ or ] indicate that an endpoint is included in the interval [brackets]
•
( or ) indicate that an endpoint is not included (parenthesis)
• ∞ and -∞ indicate the unboundeness of an interval
Example 2:
a) Write –2 ≤ x ≤ 12 using interval notation.
b) Write x > –4 using interval notation.
c) Write x < 3 or x ≥ 54 using interval notation.
Recall….
• A relation is a rule that relates two quantities A and B
• Set A of inputs is domain
• Set B of outputs is the range
• Relations expressed 4 ways:
– Verbally: The output value is 2 more than the input.
– Numerically: table or list of ordered pairs
– Graphically: on coordinate plane
– Algebraically: equation
A function is a special type of relation.
includes the following
Functions on a Graph
Example 3:
a) Determine whether the relation represents y as a function of x.
The input value x is the height of a student in inches, and the output value y is the number of books that the student owns.
b) Determine whether the table represents y as a function of x.
c) Determine whether the graph represents y as a function of x.
d) Determine whether x = 3y 2 represents y as a function of x.
Function Notation
•
•
•
•
•
f(x)
read f of x
the value of the function f at x
corresponds to the y-value for a given x-value
y = f(x)
Equation: 𝑦 = −6𝑥
Related Function: 𝑓(𝑥) = −6𝑥
Because it can represent any value in the function’s domain, x is called the independent variable.
A value in the range of f is represented by the dependent variable, y.
Example 4:
a) If f (x) = x 2 – 2x – 8, find f (3).
b) If f (x) = x 2 – 2x – 8, find f (–3d).
c) If f (x) = x2 – 2x – 8, find f (2a – 1).
Domain
When you are given a function with an unspecified domain, the implied domain is the set of all real numbers for which the expression
used to define the function is real.
Exclude values that:
• Result in a division of 0
• Take the even root of a negative number
Example 5:
a) State the domain of the function
b) State the domain of the function
c) State the domain of the function
Piecewise Function
A function that is defined using two or more equations for different intervals of the domain
Example 6:
a) FINANCE Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage.
Their evaluation yielded the following piecewise-defined function. Find the average price per square foot for a home with the square
footage of 1400 square feet.
b) Find the average price per square foot for a home with the square footage of 3200 square feet.