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

by OC533113

Department

MathematicsCourse Code

MAT135H1Professor

Alison SmithLecture

1This

**preview**shows half of the first page. to view the full**2 pages of the document.**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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