Calculus 1
Functions
Chapter 1: Functions
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Functions and Their Graphs
Combining Functions
Injections, Surjections, Bijections
Shifting and Scaling Graphs
Trigonometric Functions (Self study, not to be covered in class)
What is a function?
• A function can be represented by an equation, a graph, a numerical table, or a
verbal description
• y= f(x) ("y equals f of x").
• f: function
• x: the independent variable representing the input value of f
• y: the dependent variable or output value of
What is a function?
• DEFINITION: A function f from a set D to a set Y is a
rule that assigns a unique (single) element f(x) of Y to
each element x of D.
• The set D of all possible input values is called the
domain of the function.
• The set of all values of f(x) as x varies throughout D is
called the range of the function
Domain and Range of a function
Visual representation of functions: Graphs
Visual representation of functions: Graphs
Graphing functions
• There are many tools for graphing functions
• For a 2D graphing tool, please see:
https://www.geogebra.org/graphing
Are all relations between x and y functions?
• Vertical Line Test for a function
Piecewise defined functions
• Absolute value function:
• Another example:
Piecewise defined functions
• Greatest Integer Function
(Integer Floor Function)
• Least Integer Function
(Integer Ceiling Function)
Increasing and Decreasing functions
Even and Odd functions
Functions with linear graphs
A function of the form f(x) = mx + b
• f(x) = 0x + b=b
• f(x) = 1x + 0=x
• f(x) = mx + 0=mx
https://www.geogebra.org/graphing
Power Functions: y=xa where a is constant
If a is a positive integer then the graphs are as follows:
Power Functions: y=xa where a is constant
• If a= -1 or a=-2 than the graphs are as follows:
Power Functions: y=xa where a is constant
• If a= 1/2 or a =1/3 than the graphs are as follows:
y=x1/2 Domain and Range ?
y=x1/3 Domain and Range ?
Algebraic Functions
• Any function constructed from polynomials using algebraic operations are called algebraic functions.
Ex: All rational functions are algebraic.
Ex: y is an algebraic function of x where y3 - 9xy + x3 = 0
More Examples:
Trigonometric functions
Exponential functions: y=ax where
Logarithmic functions: =logax where
Transcendental Functions: Non algebraic functions
• Trigonometric
• Inverse trigonometric
• Exponential
• Logarithmic functions
• Hyperbolic
• ……
• and many others are all transcendental
functions
Combining Functions
Combining Functions
Combining Functions
Combining Functions
Combining Functions
Composite Functions
Inverse functions
Definition: If composition of two functions f and g is an identity function
then f and g are inverses of each other:
If 𝑓 ∘ 𝑔 𝑥 = 𝑓(𝑔(𝑥)) = 𝑥 then 𝑓 = 𝑔−1 (means inverse of g)
Example: 𝑓(𝑥) = 𝑒 𝑥 and 𝑔(𝑥) = ln(𝑥)
𝑓 ∘ 𝑔 𝑥 = 𝑓 ln 𝑥 = 𝑒 ln(𝑥) = 𝑥
𝑔 ∘ 𝑓 𝑥 = 𝑔 𝑒 𝑥 = ln(𝑒 𝑥 ) = 𝑥
Thus, 𝑓 = 𝑔−1 and 𝑔 = 𝑓 −1
Injection (One to one) , Surjections (Onto), Bijection
• A function f: A → 𝐵 is an one-to-one function (also injection) provided that
If for every x, y in 𝐴, 𝑖𝑓 𝑥 ≠ 𝑦 𝑡ℎ𝑒𝑛 𝑓(𝑥) ≠ 𝑓(𝑦)
Examples:
• 𝑓 𝑥 = 3𝑥 is an injective function from 𝑹 to 𝑹
• 𝑓 𝑥 = 𝑥 2 is not an injective function from 𝑹 to 𝑹
• 𝑓 𝑥 = 𝑥 2 is an injective function from 𝑹+ 𝑡𝑜 𝑹+
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A function f: A → 𝐵 is an onto function (also surjection) provided that
If for every y in 𝐵, 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑥 𝑖𝑛 𝐴 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑦 = 𝑓(𝑥)
Examples:
• 𝑓 𝑥 = 3𝑥 is an onto function from 𝑹 to 𝑹
• 𝑓 𝑥 = 𝑥 2 is not an onto function from 𝑹 to 𝑹
• 𝑓 𝑥 = 𝑥 2 is an onto function from 𝑹+ 𝑡𝑜 𝑹+
If a function 𝑖𝑠 𝑖𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑠𝑢𝑟𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑒𝑛 𝑖𝑡 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑏𝑖𝑗𝑒𝑐𝑡𝑜𝑛
Example: 𝑓 𝑥 = 3𝑥 is a bijection from 𝑹 to 𝑹
Shifting Graphs
Shifting Graphs
Scaling a Graph of a Function
Reflecting a Graph of a Function
Scaling and Reflecting a Graph of a Function
Scaling and Reflecting a Graph of a Function
Example
Trigonometry: A short revision (self study)
Angles
Measuring Angles
𝑠 = 𝑟𝜃
𝑠 𝑟𝜃
𝜃 𝑟𝑎𝑑 = =
𝑟
𝑟
𝑖𝑓 𝑟 = 1 𝑡ℎ𝑒𝑛 𝑠 = 𝑟𝜃 = 𝜃
𝑠 𝑟
and the measure of the angle is ∶ = = 1 𝑟𝑎𝑑𝑖𝑎𝑛
𝑟 𝑟
Angles
Radians and Degrees
Basic trigonometric functions
Basic trigonometric functions
Graphs of trigonometric functions
Deriving some trigonometric identities
Sinusoids
Sinusoids