Calculus

1. Basic Calculus
Function -A function (f) is a rule that assigns exactly one element y in
a set B to each element x in a set A.
The set of permissible values of x is called the domain of the function
and the set of all resulting value of y is called the range of the
function.
The symbol f(x) (read as “f of x”) denotes the particular value of y that
corresponds to the given value of x.
So if we write y = f(x), then x is the independent variable and y is the
dependent variable.
Given that f is the function defined by:
f(x) = 𝑥2
+ 3𝑥 − 4
Ex 1. Find: f(0), f(-2), f(3h), f(x+h), f(x)+f(h)
Ex 2. Find ((𝑥+ℎ)−𝑔(𝑥)) / ℎ where h≠0
a.) g(x) = 4𝑥2
−5𝑥+7
b.) g(x) = √(𝑥+9)
Ex 3. Let g be the function g(x)= 3x-2 if x<1
Find g(2), g(-4), g(1) x² if 1⦤x
Ex.

2. Graphing Function
EX. Graphing

3. Domain and Range by Graphing
Ex.

4. Trigonometric Functions
In mathematics, the trigonometric functions (also called circular
functions, angle functions) are functions of an angle.
They relate the angles of a triangle to the lengths of its sides.
Trigonometric functions are important in the study of triangles and
modeling periodic phenomena, among many other applications.
EX. Given
∅(x) = sin x
Find: a. ∅(π) b. ∅(5/4 π) c. ∅(2/3 π) d. ∅(- π/6)
Operations on Function
Given the two functions f and g:
(i)Sum, denoted by : f+g
defined by: (f+g)(x)=f(x)+g(x)
(ii)Difference, denoted by f-g
defined by: (f-g)(x)=f(x)-g(x)
(iii)Product, denoted by f.g
defined by: (f.g)(x)= f(x).g(x)
(iv)Quotient, denoted by f/g
defined by: (f/g) (x)= (f(x))/(g(x))
Ex. f(x) = 2x – 3 and g(x) = 𝑥2
+1
find: a.) f + g b.) f . g c.) g/f
Composite Functions
(f og)(x) = f(g(x)), the g function is inside of the f function
(g o f )(x) = g(f(x)), the f function is inside of the g function
Ex. f(x)=
5
𝑥−2
g(x)= 2x+1
Find: (f °g)(3)
Ex f(x)= 2x - 3 g(x) = cos x
Find: f °f g°g
f °g g °f