Differential Calculus
Engr. Joeydann M. Telin
Chapter 1
FUNCTIONS
Definition:
- A function from set X and Y is the set of
ordered pairs of distinct real numbers (x,y), in
which no two distinct ordered pairs have the
same first component.
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.
The variable x is called the independent
variable while the variable y is called the
dependent variable because y is usually
expressed in terms of x when their relationship is
given in the form of an equation.
In our high school days
Functions can be express by:
• Arrow diagrams
• Tables
• Graphs
• Equations
y = 𝑥2
Functions can be specified in a variety of ways.
Functions that are given by equations involving
the dependent and independent variables. For
instance, the equation
Equation in implicit form defines the dependent
variable, as a function of the independent
variable.
To evaluate this function (that is, to find the value that corresponds to a given -value), it is
convenient to isolate on the left side of the
equation.
Equation in explicit form Using as the name of
the function, you can write this equation as
Function notation
Ex 1.
Given that f is the function defined by:
f(x) = 𝑥 2 + 3𝑥 − 4
Find: a.) f(0)
e.) f(3x)
b.) f(3)
f.) f(x+h)
c.) f(h)
g.) f(x) + f(h)
d.) f(3h)
Ex 2.
Find
𝑔 𝑥+ℎ −𝑔(𝑥)
ℎ
a.) g(x) = 4𝑥 2 − 5𝑥 + 7
b.) g(x) = 𝑥 + 9
where h≠0
Assignment
For the function f defined by
evaluate each expression
Graph of a function
The graph of the function
consists of all points
(x, f(x)) where x is in the
domain of f,
note that x the directed
distance from the y -axis and
f(x) the directed distance
from the x -axis.
Equation:
y = x²
f(x) = 𝑥 − 2
g(x) = 𝑥² − 9
x2 + y2 = 25
The graphs of eight basic functions
Finding the Domain and Range of a Function
The domain of the function
is the set of all -values for which
which is the interval
To find the range,
observe that is never negative.
Ex 3.
Determine the Domain of the function g
defined by
g(x) = 𝑥(𝑥 − 2)
Using Calculator
MODE – 7 – TABLE
Start: -4
End: 7
Step: 1
A Function Defined by More than One
Equation
For the piecewise-defined function
Ex 4.
Let f be the function defined by
x-1
if x<3
f(x) =
5
if x=3
2x+1 if 3<x
determine the domain and range
Ex 5.
Let g be the function
g(x)=
3x-2 if x<1
x²
if 1⦤x
determine the domain and range
Ex 6.
The function h is defined by
h(x) =
𝑥²−9
𝑥−3
determine the domain and range
Ex 7.
Let H be the function defined by
H(x) = x + 3 if x ≠ 3
2
if x = 3
determine the domain and range
Ex 8.
The function of is defined by
f(x) =
x²
if x ≠ 2
7
if x = 2
determine the domain and range
Ex 9.
Determine the domain and range of the
function f for which
f(x)= x
f(x) =
x if x ≥ 0
-x if x < 0
Assignment
Ex 10.
Determine the domain and range of the
function f for which
f(x) = x(1 – x2)
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 11.
Given
∅(x) = sin x
Find:
a. ∅(π)
b. ∅(5/4 π)
c. ∅(2/3 π)
d. ∅(- π/6)
e. ∅(π-y)
Ex 12.
H(Y) = cos 2y – sin y
Find:
H(0)
H(-1/4 π)
H(π)
Ex 13.
b−b²
If F(b) =
1+b²
Find :
F(1/2)
F(tan x)
Assignment
Ex. 14
If f(w) =
1
𝑤
−
Find:
f(cos x)
𝑤
1+sin 𝑥
Operation on Functions
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))
Example 1
f(x) = 2x - 3
g(x) = 𝑥 2 +1
find:
a.) f + g
b.) f g
c.) g/f
(v)Composite, denoted by f °g
Defined by:
(f °g)(x)=f(g(x))
Example 2
f(x)=
5
𝑥−2
g(x)= 2x+1
Find: (f °g)(3)
in two methods
Example 3
f(x)= 2x - 3
g(x) = cos x
Find:
f °f
g°g
f °g
g °f
Assignment
Example 4
f(x)= sin x
g(x) = π x
Find:
(f °g) (1/2)
(g °f) (π /4)
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