8.1 Introduction The temperature at which water boils depends on - - PDF document

Chapter 8: Functions and Their Graphs

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SET 2

Chapter 8

Functions and Their Graphs

لاةـيناـيبلا اـهموسر و لاودـ

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Chapter 8: Functions and Their Graphs

8.1 Introduction ةـمذـقم  The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend).  The interest paid on a cash investment depends on the length of time the investment is held.  The area of a circle depends on the radius of the circle.  The distance an object travels from an initial location along a straight line path depends on its speed.  In each of the previous cases, the value of one variable quantity, which we might call y, depends on the value of another variable quantity, which we might call x.  Since the value of y is completely determined by the value of x, we say that: y is a function of x.  Often the value of y is given by a rule or formula that says how to calculate it from the variable x.  For instance, the equation A = πr2 is a rule that calculates the area A of a circle from its radius r.  A symbolic way to say “y is a function of x” is by writing. y = f (x) (“y equals f of x”)  In this notation:

• The symbol ƒ represents the function.
• The letter x, called the independent variable, represents the input value of ƒ.
• The letter y, called the dependent variable, represents the corresponding
• utput value of ƒ at x.

Definition of Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) Y to each element x D.

Chapter 8: Functions and Their Graphs

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8.2 Domain and Range لـباـقملا لاـجملا و لاـجملا  The set D of all possible input values is called the domain of the function.  The set of all values of ƒ(x) as x varies throughout D is called the range of the

• function. The range may not include every element in the set Y.

 Think of a function ƒ as a kind of machine that produces an output value ƒ(x) in its range whenever we feed it an input value x from its domain, and as shown if the figure below.  A function can also be pictured as an arrow diagram. Each arrow associates an element of the domain D to a unique or single element in the set Y.  In the figure below, the arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on.

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Chapter 8: Functions and Their Graphs

Example 1. Find the domain and the range of the functions:

(a) y = x2 (b) y = 1/x (c) x y  (d) x y   4 (e)

2

1 x y  

Solution: (a) The formula y = x2 gives a real y-value for any real number x, so the domain is (– ∞, ∞). The range of y = x2 is [0, ∞) because the square of any real number is nonnegative and every nonnegative number y is the square of its own square root,

 2

y y 

for y ≥ 0. (b) The formula y = 1/x gives a real y-value for every x except x = 0. We cannot divide any number by zero. The range of y = 1/x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y = 1/(1/y). (c) The formula

x y 

gives a real y-value only if x ≥ 0. The range of

x y 

is [0, ∞) because every nonnegative number is some number’s square root (namely, it is the square root of its own square). (d) In

x y   4

, the quantity 4 – x cannot be negative. That is, 4 – x ≥ 0, or x ≤ 4. The formula gives real y-values for all x ≤ 4. The range of

x y   4

is [0, ∞), the set of all nonnegative numbers. (e) The formula

2

1 x y  

gives a real y-value for every x in the closed interval from – 1 to 1. Outside this domain, 1 – x2 is negative and its square root is not a real number. The values of 1 – x2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of

2

1 x y  

is [0, 1]. he answer of example 1 is summarized in the table below.

Chapter 8: Functions and Their Graphs

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Example 2. If

12 7 6 ) (

2

   x x x f

, find: (a) ) 5 ( f (b) ) 4 ( f (c) ) (b f (d) ) 2 ( a f 

Solution:

(a) 12 ) 5 ( 7 ) 5 ( 6 ) 5 (

2

   f 127 12 35 150 12 35 ) 25 ( 6        (b) 12 ) 4 ( 7 ) 4 ( 6 ) 4 (

2

      f 136 40 96 12 28 ) 16 ( 6       (c) 12 ) ( 7 ) ( 6 ) (

2

   b b b f 12 7 6 2    b b (d) 12 ) 2 ( 7 ) 2 ( 6 ) 2 (

2

      a a a f 22 17 6 7 2 6 24 24 12 7 14 ) 4 4 ( 6

2 2 2

              a a a a a a a a

8.3 Graphs of Functions لاوذلل يناـيبلا مسرـلا  If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ.  In set notation, the graph is: {(x, f(x)) | x  D}  The graph of the function f(x) = x + 2 is the set of points with coordinates (x, y) for which y = x + 2. Its graph is sketched in the figure below.

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Chapter 8: Functions and Their Graphs

8.4 The Vertical Line Test يدوـمعلا طـخلا راـبتخإ  A function ƒ can have only one value ƒ(x) for each x in its domain.  No vertical line can intersect the graph of a function more than once.  Thus, a circle cannot be the graph of a function since some vertical lines intersect the circle twice (see figure a below).  If a is in the domain of a function ƒ, then the vertical line x = a will intersect the graph of ƒ in the single point (a, ƒ(a)).  The circle in figure a below, however, does contain the graphs of two functions of x; the upper semicircle defined by the function:

2

1 ) ( x x f  

and the lower semicircle defined by the function

2

1 ) ( x x g    (see figures b and c below).

Chapter 8: Functions and Their Graphs

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Example 3. Which of the graphs below are graphs of functions of x, and which are not? Solution: Only graph b is a graph of a function of x. 8.5 Even Functions and Odd Functions ةـيدرـفلا لاوذـلا و ةـيجوزـلا لاوذـلا Definition: Even Function and Odd Function A function y = ƒ(x) is an: even function of x if ƒ( x) = ƒ(x), or

• dd function of x if ƒ( x) =  ƒ(x),

for every x in the function’s domain.  The graph of an even function is symmetric about the y-axis.

 The graph of an odd function is symmetric about the origin. Even function (symmetric about the y-axis)

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Chapter 8: Functions and Their Graphs

Odd function (symmetric about the origin)

Example 4. Determine whether the following functions are even, odd, or neither:

(a)

2

7 ) ( x x f  (b)

6 4 ) (   x x g

(c)

3

6 ) ( x x x h   (d) 6 4 ) (   x x F

Solution:

(a)

2 2

7 ) ( 7 ) ( x x x f     . Since ƒ( x) = ƒ(x), f is an even function. (b)

6 4 6 ) ( 4 ) (        x x x g

. Also,

6 4 ) 6 4 ( ) (        x x x g

. Since neither g( x) = g(x) nor g( x) =  g(x) is the case, the function g is neither even nor odd. (c)

3 3

6 ) ( 6 ) ( x x x x x h         . Thus, h( x) =  h(x) and h is an odd function. (d) 6 4 6 4 ) (        x x x F . Since neither F( x) = F(x) nor F( x) =  F(x) is the case, the function F is neither even nor odd.

8.6 Linear Functions لاوذلا ةـيطخلا  A function of the form f(x) = mx + b, for constants m and b, is called a linear function.  The figure below shows an array of lines f(x) = mx Where b = 0, so these lines pass through the origin.

Chapter 8: Functions and Their Graphs

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 Constant functions result when the slope m = 0 and as shown below. Example 5. The cost of Omantel home internet service is OMR 20 per month with download limit of 50 GB. The cost of data downloading beyond the limit is one Rial for each extra downloaded GB. (a) Write the cost function of this service. (b) Graph the function using the following values

• f downloaded data:

{50, 55, 60, 65, 70, 75}. (c) Algebraically, find the cost when the total downloaded data is 70 GB. (d) From the graph, find how much downloaded data cost OMR 35. Solution: (a) Assume x is the total downloaded data in GB. The the cost function in OMR is:

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Chapter 8: Functions and Their Graphs

f (x) = 20 + 1 (x – 50) f (x) = 20 + x – 50 f (x) = x – 30 (b) The graph of the function is as shown below: (c) f (x) = x – 30 f (70) = 70 ‒ 30 = OMR 40 (d) From the graph of the function, the downloaded data that costs OMR 35 is 65 GB. 8.7 Quadratic Functions ةـيـعيـبرتـلا لاوذـلا

 The function given by f (x) = ax2 + bx + c is called

a quadratic function. Where a, b, and c be real numbers with a = 0.  Quadratic functions are second degree polynomial functions.  The basic quadratic functions are the functions f (x) = x2 and f (x) =  x2. The graph of each of these two functions is a parabola with vertex at the origin (0,0), and the axis of symmetry is the y-axis and as shown in the figure below. Data (GB) 50 55 60 65 70 75 Cost (OMR) 20 25 30 35 40 45

20 25 30 35 40 45 45 50 55 60 65 70 75

Cost (OMR) Downloaded Data (GB) f(x) = x − 30

Chapter 8: Functions and Their Graphs

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• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6

• 4
• 3
• 2
• 1

1 2 3 4

y x

f(x) = x2 − x − 6 Zeros

 The graph of a quadratic function is a parabola which can cross

the x-axis once, twice, or never.

 These points of intersection are called x-intercepts or zeros, and

they may be found algebraically or graphically.

 The algebraic methods used for finding the zeros of quadratic

functions are: factorisation, quadratic formula, and completing the square.

 For example, the zeros of the quadratic function

f(x) = x2 – x – 6 = 0 are x = 3 and x = – 2 and as shown in the figure below.

 The standard form of quadratic functions is

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Chapter 8: Functions and Their Graphs

f (x) = a(x – h)2 + k, a ≠ 0.

 The graph of f(x) = a(x – h)2 + k is a parabola whose axis is the vertical line x = h and whose vertex is the point (h, k).  If a > 0, the parabola opens upward.  If a < 0, the parabola opens downward.  For positive a, the quadratic function f(x) = a(x – h)2 + k has a minimum value of k. This value is attained when x = h.  For negative a, the quadratic function f(x) = a(x – h)2 + k has a maximum value of k. This value, also, is attained when x = h. The maximum and minimum values of quadratic functions may be found as follows:

• 1. If a > 0, f has a minimum at

a b x 2  

.

• 2. If a < 0, f has a maximum at

a b x 2  

. Example 6. Graph the function y = x 2 over the interval [–2, 2] Solution:

• 1. Make a table of xy-pairs that satisfy the function

rule, in this case the equation y = x2 .

• 2. Plot the points (x, y) whose coordinates

appear in the table.

Chapter 8: Functions and Their Graphs

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• 3. Draw a smooth curve through the plotted
• points. Label the curve with its equation.

Example 7. For the function f(x) = x2 + 6x + 8:

(a) draw the graph using values of x = {– 6, – 5, – 4, – 3, – 2, – 1, 0}. (b) find the zeros graphically. (c) find the minimum value graphically. (d) find the zeros algebraically. (e) find the minimum value algebraically. Solution: (a)

x

 6  5  4  3  2  1

f(x)

8 3  1 3 8

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Chapter 8: Functions and Their Graphs

• 2
• 1

1 2 3 4 5 6 7 8 9

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2

y x

f(x) = x2 + 6x + 8

minimum

(b) The graph intersects the x-axis at x =  4 and x =  2 which are the zeros of f(x) = x2 + 6x + 8. (c) From the graph, the function has a minimum value of  1. This minimum value is attained when x =  3. This means the vertex of f(x) = x2 + 6x + 8 is ( 3,  1). (d) To find the zeros of f(x) = x2 + 6x + 8 algebraically, set f(x) = 0 and solve for x using any of the three algebraic methods used for solving quadratic equations. Thus, and using factorasation: f(x) = x2 + 6x + 8 = 0 (x + 2)(x + 4) = 0 Either (x + 2) = 0 which gives x =  2,

• r (x  4) = 0 which gives x =  4.

Thus, the zeros of f(x) = x2 + 6x + 8 are x =  2 and x =  4

which are the same as those found from the graph.

(e) The function f(x) = x2 + 6x + 8 has a minimum at a b x 2   . 3 ) 1 ( 2 6 2       a b x Minimum value = f(– 3) = (– 3)2 + 6(– 3)+ 8 = – 1 Thus, f(x) = x2 + 6x + 8 has a vertex at (– 3, – 1), i.e. it has a minimum value = – 1 at x = – 3.

Chapter 8: Functions and Their Graphs

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This result matches the result found graphically in (c).

Example 8. For the function f(x) = – x2 + 6x – 8 :

(a) draw the graph using values of x = {0, 1, 2, 3, 4, 5, 6}. (b) find the maximum value algebraically. Solution: (a) (b) The function f(x) = – x2 + 6x – 8 has a maximum at

a b x 2  

.

3 ) 1 ( 2 6 2       a b x

Maximum value = f(3) = – (3)2 + 6(3) – 8 = 1 Thus, f(x) = – x2 + 6x – 8 has a vertex at (3, 1), i.e. it has a maximum value = 1 at x = 3. This result can be easily recognized from the graph of the function.

x

1 2 3 4 5 6

f(x)

 8  3 1  3  8

• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2

• 2
• 1

1 2 3 4 5 6 7

y x

f(x) = − x2 + 6x − 8

maximum

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Chapter 8: Functions and Their Graphs

Example 9. A baseball is hit at a point 3 feet above the ground at a

velocity of 100 feet per second and at an angle of 45º with respect to the ground. The path of the baseball is given by the function f(x) = – 0.0032x2 + x + 3, where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball? Solution: For this quadratic function, we have

3 0032 . ) (

2

    x x x f

Which implies that a = – 0.0032 and b = 1. Because the function has a maximum when

a b x 2  

, we can conclude that the baseball reaches its maximum height when it is x feet from home plate, where x is

a b x 2   25 . 156 ) 0032 . ( 2 1    

feet. At this distance, the maximum height is

3 25 . 156 ) 25 . 156 ( 0032 . ) 25 . 156 (

2

    f

= 81.125 feet 8.8 Exponential Functions: لاوذـلا ةـيــسلؤا  Functions of the form f(x) = a x, where the base a > 0 is a positive constant and a ≠ 1, are called exponential functions.  All exponential functions have domain (– ∞, ∞) and range (0, ∞).  The graphs of some exponential functions are shown below.

Chapter 8: Functions and Their Graphs

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8.9 Logarithmic Functions ةيـمتراـغوـللا لاوذـلا

 These are the functions f(x) = log a x, where the base

a ≠ 1 is a positive constant.

 The figure below shows the graphs of four logarithmic

functions with various bases.