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Lecture 1

MAT135H1 Lecture Notes - Lecture 1: Piecewise, Real Number, Dependent And Independent Variables


Department
Mathematics
Course Code
MAT135H1
Professor
Alison Smith
Lecture
1

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MAT135 - Lecture 1 - Functions, Finding Domains, Vertical Line Test, Piecewise Defined
Functions
Introduction to course
Note that the official website for this course is www.math.utoronto.ca/lam (the course is NOT
using Blackboard/Portal)
Print out the 8-page Course Outline from the website.
Course administrator is Anthony Lam. His office is BA6125. He can be contacted at:
lam@math.utoronto.ca.!
- only use your mail.utoronto.ca email account to contact him!
- the subject line of your email should contain the words “MAT135”
Functions
Function (f) is a rule that assigns to each element x of a set D exactly one element called f(x)
in a set E.!
- normally, D and E are sets of real numbers, in this case a function takes a number, x, and
produces a number, f(x).!
- D is the domain of the function (set of inputs into the function).!
- Range = set of all values taken by f(x)!
- If we write y=f(x), y is called the dependent variable, x is the independent variable.
Example: f(x) = x2 !
- Domain of the function is R, or (-, ), i.e. all Real Numbers.!
- Range of function must be positive (the square of any x is greater than or equal to zero, and
any positive number is in fact a square), so the range of f is [0, ).
Examples of Finding Domain
f(x) = (3+x) / (x(x+1))!
- Any Real Number x is in the domain unless the denominator = 0 (x(x+1)=0)!
- Therefore the Domain: (-, -1) U (-1, 0) U (0, ), where U denotes “union,” meaning “and.”
f(x) = (5-x)!
- What can go wrong here? Remember:!
1. Can only take of positive numbers and zero.!
2. For x to be in domain, x must be greater than or equal to zero so that x makes sense!
3. Also need (5-x) 0. Rearranging, 5 x, so x 25.!
- Therefore the Domain is [0, 25].
Vertical Line Test
A curve in the xy plane is the graph of a function of x if an d only if no vertical line intersects
the curve more than once.
Piecewise Defined Functions
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