Functions
—————————–
Phan Thi Khanh Van
E-mail: khanhvanphan@hcmut.edu.vn
October 22, 2020
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Functions
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Table of contents
1
Functions
2
Power Functions
3
Exponential Functions
4
Logarithmic Functions
5
Trigonometric Functions
6
New Functions from Old Functions
7
Composite function
8
Inverse function
9
Inverse trigonometric functions
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Functions
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Functions
Functions
A function f is a rule that
assigns to each element x
in a set D exactly one
element, called f (x), in a
set E .
The set D is called the domain of the function.
The number f (x) is the value of f at x.
The range of f is the set of all possible values of f (x) as
x varies through-out the domain:
R = {y ∈ E | ∃x ∈ D : y = f (x)}
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Functions
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Functions
Example 1 (algebraically)
The amount of pollution in a certain lake is found to be
1
A(t) = (t 4 + 3)3
where t is measured in years and A(t) is measured in
appropriate units. Find the domain and range of A.
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Functions
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Functions
Example 2 (verbally)
A rectangular storage container with an open top has a
volume of 10m3 . The length of its base is twice its width.
Material for the base costs $10 per square meter; material
for the sides costs $6 per square meter. Express the cost
C of materials as a function of the width of the base.
Find the domain and range of C .
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Functions
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Functions
Example 3 (visually)
The graph of a function f
is shown in the figure. Find
the domain and range of f .
Find f (1).
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Functions
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Functions
Example 4 (numerically)
Temperature readings T (in o F ) were recorded every two
hours from midnight to 2 : 00 PM in Phoenix on
September 10, 2008. The time t was measured in hours
from midnight. Find T (10). Find the domain and range
of T .
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Functions
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Functions
Parametric function
(
x = x(t)
y = y (t)
Example
(
x = 1 + 3 cos(t)
The parametric curve
is the circle of
y = 2 + 3 sin t
radius 3 and center I (1; 2): (x − 1)2 + (y − 2)2 = 9
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Functions
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Functions
Piecewise Defined Functions
A cell phone plan has a basic charge of $35 a month.
The plan includes 400 free minutes and charges 10 cents
for each additional minute of usage. Write the monthly
cost C as a function of the number x of minutes used
and graph C as a function of x for 0 ≤ x ≤ 600.
(
35, 0 ≤ x ≤ 400,
C (x) =
35 + 0.1(x − 400), 400 < x ≤ 600.
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Functions
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Functions
Symmetry
Even function
f (x) = f (−x)
Odd function
f (x) = −f (−x)
f (x) = x 3 : (−x)3 = −x 3 : odd function,
f (x) = cos x: cos(−x) = cos x: even function
f (x) = 1 + x 3 : neither odd nor even.
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Functions
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Functions
Increasing and Decreasing Functions
Increasing function
Decreasing function
x1 < x2 ⇒ f (x1 ) < f (x2 )
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Functions
x1 < x2 ⇒ f (x1 ) > f (x2 )
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Power Functions
Power Functions x α , α = 2k
*
D = R.
*
R = [0, ∞).
*
Increasing in (0, ∞), decreasing in(−∞, 0).
*
Even function.
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Functions
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Power Functions
Example
The formula for the horsepower rating H of an engine
with N cylinders is
ND 2
H(D) =
2.5
where D is the diameter of a cylinder.
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Functions
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Power Functions
Power Functions x α , α = 2k + 1
*
D = R.
*
R = R.
*
Increasing in R.
*
Odd function.
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Functions
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Power Functions
Power Functions x α = x1 , α = −1
*
*
*
*
D = R\{0}.
R = R\{0}.
Decreasing in (0, ∞), and (−∞, 0).
Odd function.
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Functions
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Power Functions
1
Power Function y = x 2 =
*
*
*
*
√
x, α =
1
2
D = [0, ∞).
R = [0, ∞).
Increasing in [0, ∞).
Neither odd nor even function.
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Functions
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Exponential Functions
Exponential Functions y = ax
*
Domain: D = R.
*
Range: R = (0, ∞).
*
Increasing in R
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Functions
(a > 1)
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Exponential Functions
Exponential Functions y = ax
*
Domain: D = R.
*
Range: R = (0, ∞).
*
Decreasing in R
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Functions
(0 < a < 1)
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Exponential Functions
Formula
1
2
3
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am .an = am+n
4
(am )n = amn
5
am .b m =
(ab)m
6
Functions
a−m =
1
am
am
= am−n
n
a
a n a n
=
bn
b
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Exponential Functions
Drug Medication
The function
D(h) = 5e −0.4h
can be used to find the number of milligrams D of a
certain drug that is in a patient’s bloodstream h hours
after the drug has been administered. How many
milligrams will be present after 1 hour? After 6 hours?
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Functions
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Logarithmic Functions
Logarithmic Functions y = loga x, a > 1
*
Domain: D = (0; +∞).
*
Range: T = R.
*
Increasing function if a > 1, decreasing if a < 1.
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Functions
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Logarithmic Functions
Formual
i)
ln(xy ) = ln x + ln y
ii)
ln x α = α ln x
iii)
e ln x = x
a)
b)
c)
ln e = 1
ln 1 = 0
1
ln 2 = −2
e
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d)
e)
Functions
ln(−1) =?
ln? = −1
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Logarithmic Functions
Chemistry
The pH of a chemical solution is given by the formula
pH = − log10 [H + ]
where [H + ] is the concentration of hydrogen ions in
moles per liter. Values of pH range from 0 (acidic) to 14
(alkaline). Distilled water has a pH of 7.
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Functions
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Trigonometric Functions
Trigonometric Function y = sin x
*
Domain: D = R.
*
Range: R = [−1, 1].
*
Periodic function, period: 2π
sin(x) = sin(x + 2π)
*
Odd function
2
y
y = sin x
1
−2 ∗ pi
−pi
0
−pi/2
pi/2
pi
3 ∗ pi/2 2 ∗ pi 5 ∗ pi/2
x
−1
−2
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Functions
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Trigonometric Functions
Formula
i)
sin(π − x) = sin(x), sin(−x) = − sin x
ii)
sin2 x + cos 2 x = 1
iii)
sin 2x = 2 sin x cos x
1 − cos 2x
iv)
sin2 x =
2
a.
b.
sin 0 = 0
√
π
2
sin =
4
2
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c.
d.
Functions
5π
π
1
= sin =
6 √ 6
2
3
sin? =
2
sin
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Trigonometric Functions
Example
A Cepheid variable star is a star whose brightness
alternately increases and decreases. The most easily
visible such star is Delta Cephei, for which the interval
between times of maximum brightness is 5.4 days. The
average brightness of this star is 4.0 and its brightness
changes by ±0.35. The brightness of Delta Cephei at
time t, where t is measured in days, has been modeled by
the function
2πt
B(t) = 4 + 0.35 sin(
).
5.4
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Functions
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Trigonometric Functions
Trigonometric function y = cos x
*
Domain: D = R.
*
Range: R = [−1, 1].
*
Periodic function, period: 2π
cos(x) = cos(x + 2π)
*
Even function
2
y
y = cos x
1
−2 ∗ pi
−pi
0
−pi/2
pi/2
pi
3 ∗ pi/2 2 ∗ pi 5 ∗ pi/2
x
−1
−2
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Functions
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Trigonometric Functions
Formula
i)
cos(π − x) = − cos x, cos(−x) = cos x
ii)
cos 2x = cos2 x − sin2 x
iii)
cos 2x = 2 cos2 x − 1 = 1 − 2 sin2 x
1 + cos 2x
iv)
cos2 x =
2
π
v)
cos( + kπ) = 0, ∀k ∈ Z
2
a.
cos
±π
=0
2
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b.
Functions
cos
2π
π
1
= − cos = −
3
3
2
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Trigonometric Functions
Trigonometric function y = tan x
*
Domain: D = R\{ π2 + kπ}.
*
Range: R = R.
*
Periodic function, period: π
tan(x) = tan(x + π)
*
Odd function
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Functions
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Trigonometric Functions
y
y = tan x
x
−pi
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−pi/2
0
Functions
pi/2
pi
3 ∗ pi/2
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Trigonometric Functions
Formula
i)
ii)
iii)
iv)
sin x
cos x
tan(π − x) = tan(−x) = − tan x
tan(π + x) = tan(x)
tan 0 = 0, tan( π2 ) is not defined
tan x =
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Functions
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New Functions from Old Functions
Translating the graph of f
Vertical and Horizontal Shifts
Suppose that c > 0,
*
y = f (x) + c, shift the graph of
c units upward.
*
y = f (x) − c, shift the graph of
c units downward
*
y = f (x − c), shift the graph of
c units to the right
*
y = f (x + c), shift the graph of
c units to the left
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Functions
y = f (x) a distance
y = f (x) a distance
y = f (x) a distance
y = f (x) a distance
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New Functions from Old Functions
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Functions
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New Functions from Old Functions
Stretching the graph of f
Vertical and Horizontal Stretching
Suppose that c > 1,
*
y = cf (x), stretch the graph of y = f (x) vertically by
a factor of c
*
y = c1 f (x), shrink the graph of y = f (x) vertically by
a factor of c
*
y = f (cx), shrink the graph of y = f (x) horizontally
by a factor of c
*
y = f (x/c), stretch the graph of y = f (x)
horizontally by a factor of c
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Functions
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New Functions from Old Functions
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Functions
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New Functions from Old Functions
Reflecting the graph of f
Vertical and Horizontal Reflecting
*
y = −f (x), reflect the
graph of y = f (x)
about the x-axis
*
y = f (−x), reflect the
graph of y = f (x)
about the y-axis
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Functions
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Composite function
Composite function
Given two functions f and
g , the composite function
f ◦ g (also called the
composition of f and g ) is
defined by
f ◦ g (x) = f (g (x))
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Functions
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Composite function
Example
1
2
√
f = x 2 + 1, g = x +√1
f ◦ g = x + 2, g ◦ f = x 2 + 2
Domain of g ◦ f : D = R.
Domain of f ◦ g : D = [−1, ∞)
If you invest x dollars at 4% interest compounded
annually, then the amount A(x) of the investment
after one year is A(x) = 1.04x. Find A ◦ A, A ◦ A ◦ A,
A ◦ A ◦ A ◦ A. What do these compositions represent?
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Functions
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Composite function
Example
A ship is moving at a speed of 30km/h parallel to a
straight shoreline. The ship is 6km from shore and it
passes a light- house at noon.
a)
Express the distance s between the lighthouse and the
ship as a function of d, the distance the ship has
traveled since noon; that is, find f so that s = f (d).
b)
Express d as a function of t, the time elapsed since
noon; that is, find g so that d = g (t).
c)
Find f ◦ g . What does this function represent?
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Functions
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Inverse function
One-to-one Function
y = f (x) is called a one to one function if for every
y ∈ R, there exists one and only one x ∈ D such that
y = f (x).
One-to-one function
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Not one-to-one function
Functions
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Inverse function
Inverse function
Let y = f (x) be a one-to-one function. We have
∀y ∈ R, ∃!x ∈ D : x = g (y ).
y = g (x) = f −1 (x) is called the inverse function of f (x).
f (f −1 (y )) = y ,
f −1 (f (x)) = x
Example
√
1
Given y = f (x) = x 3 ⇔ x = 3 y
√
Then, f −1 (x) = 3 x.
√
3
x 3 +1
2
Given y = f (x) =
e
⇔
x
=
ln y − 1
√
3
−1
Then, f (x) = ln x − 1.
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Functions
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Inverse function
The graph of f and f −1 is symmetric about the line y =
x
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Functions
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Inverse trigonometric functions
Inverse trigonometric function y = arcsin x
y = arcsin x
⇐⇒ x = sin y
π
π
D : −1 ≤ x ≤ 1
R :− ≤y ≤
2
2
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Functions
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Inverse trigonometric functions
Inverse trigonometric function y = arccos x
y = arccos x
⇐⇒ x = cos y
D : −1 ≤ x ≤ 1,
R :0≤y ≤π
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Functions
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Inverse trigonometric functions
Inverse trigonometric function y = arctan x
y = arctan x ⇐⇒ x = tan y
π
π
D : R,
R :− <y <
2
2
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Functions
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Inverse trigonometric functions
Inverse trigonometric function y = arccot x
y = arccot x ⇐⇒ x = cot y
D : R,
R :0<y <π
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Functions
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Inverse trigonometric functions
Properties
arcsin x + arccos x =
π
2
arctan x + arccot x =
π
2
(
arctan x +
(Phan Thi Khanh Van)
arctan x1
=
Functions
π
2, x > 0
− π2 , x <
0
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Inverse trigonometric functions
Example
a)
arcsin 1 = π2
√
2
b)
arccos
= π4
√2
c)
arctan 3 = π3
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d)
e)
f)
Functions
arctan(−1) = − π4
√
arcsin 3 =?
arcsin? = −2
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Inverse trigonometric functions
Thank you for your attention!
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Functions
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