Functions
Relations
Functions
Algebraic Test for Functions
Vertical Line Test
Function Notation
x is the input, y is the output.
Domain and Range
Rational Functions
Linear and Quadratic functions are members of Polynomials, which can all be written
in the form:
A Rational Function is a function such that one polynomial is divided by another
polynomial.
Rational Functions: Reciprocal Functions
Consider the function and its graph 2 asymptotes:
Rational Functions: y = (ax + b) / (cx + d), c does not equal to 0
Vertical and Horizontal Asymptote
From the previous 2 kinds of Rational Functions, we can see that:
Vertical Asymptote is obtained when setting the Denominator = 0.
Horizontal Asymptote is obtained by
-
First divide both the numerator and denominator by x
Cancel out x whenever possible for individual sub-fractions
Sub-Fractions with a denominator x is equal to 0
Composite Functions
X is the 1st input, g(x) is the 1st output and the 2nd input, f(g(x)) is the final output.
Inverse Functions
+ and – are inverse of each other, x and / are inverse of each other.
y = ax + b can be inverted as x = (y – b) / a
Since we want to denote the input as x and output as y, we interchange the notation
of x and y in the inverted function, so it becomes
y = (x – b) / a
One-to-One and Many-to-One Functions
Eg. 𝑦 = 𝑥 3 is a one-to-one function, as each value of x can produce one value of y. It
will pass both of vertical line test and horizontal line test.
Eg. 𝑦 = 𝑥 2 is a many-to-one function, as 2 values of x can produce same value of y.
It will pass the vertical line test (hence it is a function), but not the horizontal line test
(hence the inverse of it cannot pass the vertical line test).
Note, however if we define the domain 𝑦 = 𝑥 2 as x > 0, then it becomes a one-to-one
function.
Invertibility
Whether a function (defined as passing the vertical line test) can pass the horizontal
line test determines whether it is invertible.
Properties of Inverse Functions
Self-Inverse Functions