MAT135H1 Lecture Notes - Lecture 1: Piecewise, Real Number, Dependent And Independent Variables
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MAT135 - Lecture 1 - Functions, Finding Domains, Vertical Line Test, Piecewise Defined
Introduction to course
•Note that the official website for this course is www.math.utoronto.ca/lam (the course is NOT
•Print out the 8-page Course Outline from the website.
•Course administrator is Anthony Lam. His office is BA6125. He can be contacted at:
- only use your mail.utoronto.ca email account to contact him!
- the subject line of your email should contain the words “MAT135”
•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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