Section 1.1 Functions
Objective: In this lesson you will learn how to evaluate functions and find their domains.
IA. Interval Notation:
Inequalities can be written in a “shorthand” form called Interval notation.
ReWrite the following using Interval Notation:
1) -8 < x £ 16 2) x < 11 3) x £ -16 or x > 5
Practice: 1) -4 £ y < -1 2) a ³ -3 3) x > 9 or x < -2
IB. Intro to functions
Function A function f 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.
Domain The set of inputs of the function f.
Range The set of all outputs for the given set of inputs of the function f.
Example 1: S = {(1, 3), (-5, 4), (7, 3)}
a) What is the domain of S? b) What is the range of S? c) Is S a function? Why or why not?
Example 2. Which of the following equations represents y as a function of x? (ie: solve for y and see if it is unique)
a) 2x2 + y + 1 = 0
b) x + y2 − 6 = 0
Summary any equation where the exponent of y is even or |y| then it is not a function.
Example 3. Vertical line test:
II. Function Notation
The symbol f(x) is function notation for the value of f at x or f of x.
f is the name of the function. It can be named using any letter.
f(x) is the output value (y) of the function at the input value x.
Which means y = f(x).
If y = x + 3, then we can say f(x) = x + 3.
If y = p + 3, then we can say g(p) = p + 3.
Example: If x = m - 3, then write f(x) = x + 6 as a function of m.
Example: Write the area of a circle as a function of its circumference.
Example: Soft drink can ratio h = 4
a) express as a function of radius b) express as function of height
Example : If f (w) = 4w3 - 5w2 - 7w +13 , describe how to find f (-2) .
Example 2. If f(x) = x2 – 4x , find the following.
a) f(3) = b) f(a) = c) f(x+h) =
A piecewise-defined function is a function that is defined by two or more equations over a specified domain.
Example 3. If f(x)= x+1, if x≥0
−x, if x<0
find the following.
a) f(2) = b) f(-5) =
III. Difference Quotients:
Expression from calculus that computes slope of the secant line through 2 pts on a graph.
A difference quotient is defined as . . .
, h ¹ 0.
Example 1:
If f(x) = x2 + x − 1, find
Example 2:
If f(x) = x2 - 4x + 7, find
IV. The Domain of a Function:
Find Domain algebraically: Rules
1) If fraction then set denominator = 0 and solve
2) If an even root then set ≥ 0
3) If fraction with even root in denominator then set > 0
EVERYTHING else is “All Real Numbers”
Examples Find the domain of each of the following functions.
a) f (x) =
b) f(x) = x3 + 3x + 1
c) f(x) =
d) f(x) =
e) f(x) =