Chapter 1
Functions
By: Madam Roslina Binti Ab Ghapar
Topic 1.1
Functions and Their Graphs
Functions
Value of variable y depends on value of variable x,
𝑦=𝑓 𝑥
, we say "𝑦 equals 𝑓 of 𝑥”
Notation:
 f represents the function
 x is the independent variable representing the input
value of f
 y is the dependent variable or output value of f at x.
 D : Domain of the function. Set of possible
values of the independent variable (x).
 f(x) : Range of the function. Set of possible
resulting values of the dependent variable (y).
Domain Rules
No. Function
Domain (x)
1.
Positive power functions,
𝑓 𝑥 = 𝑥 𝑎 , 𝑎 = positive integer
Example: 𝑦 = 𝑥, 𝑦 = 𝑥 2 , 𝑦 = 𝑥 3 , …
−∞, ∞
2.
Square root/ even root 𝑓 𝑥 = 𝑏
𝑏≥0
3.
Cube root/ odd root 𝑓 𝑥 =
4.
Fractions:
a) 𝑓 𝑥 =
b) 𝑓 𝑥 =
c) 𝑓 𝑥 =
3
𝑏
−∞, ∞
(with constant numerator)
𝑐
𝑏
,
𝑐
𝑏
𝑐
3
𝑏
b any +ve power functions
𝑏≠0
𝑏>0
𝑏≠0
Example 1:
Find the domain and range for the given function.
Exercise:
Find the domain and range for the given function.
(a)
𝑓 𝑥 = 2𝑥 + 3
(e)
𝑓 𝑥 = 6 − 3𝑥
(b)
𝑓 𝑥 = 4 − 𝑥2
(f)
𝑓 𝑥 =
(c)
𝑓 𝑥 = 𝑥 2 + 2𝑥
(g)
(d)
𝑓 𝑥 = 2𝑥 + 8
(h)
𝑓 𝑥 =
𝑓 𝑥 =
𝑥 2 − 4𝑥
2
𝑥−5
4
𝑥2 − 9
Graphs of functions
Example 2:
Graph the function y = x2 over the interval [-2, 2].
The vertical line test for a function
A function can only have 1 value f(x) for each x in its
domain
Piecewise-defined functions
• Function is described by using different formulas on
different parts of its domain.
Example 3: Absolute value function
y= 𝑥 =
𝑥, 𝑥 ≥ 0
−𝑥, 𝑥 < 0
Example 4:
Sketch the graph of the function.
−𝑥,
𝑥<0
𝑓 𝑥 = 𝑥2,
0≤𝑥≤1
1,
𝑥>1
Increasing and decreasing functions
increasing
decreasing
Even functions and odd functions: symmetry
A function 𝑦 = 𝑓(𝑥) is an
Even function of x
Odd function of x
 If 𝑓(−𝑥) = 𝑓(𝑥)
 The graph is symmetry
about y-axis.
 If 𝑓 −𝑥 = −𝑓(𝑥)
 The graph is symmetry
about origin.
Example 5:
Determine whether the function is even, odd or neither.
a)
b)
c)
d)
𝑓 𝑥 = 𝑥2
𝑓 𝑥 = 𝑥2 + 1
𝑓 𝑥 =𝑥
𝑓 𝑥 =𝑥+1
Not odd/even
function
Both are
even
functions
Odd function
Common functions
1) Linear functions
f(x)= mx + c
2) Power functions 𝑓 𝑥 = 𝑥 𝑎
(a) 𝑎 = 𝑛, positive integer
2) Power functions 𝑓 𝑥 = 𝑥 𝑎
(b) 𝑎 = −1 or 𝑎 = −2
2) Power functions 𝑓 𝑥 = 𝑥 𝑎
(c) 𝑎 =
1 1 3 2
, , ,
2 3 2 3