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1. FUNCTIONS AND MODELS. FUNCTIONS AND MODELS. 1.3 New Functions from Old Functions. In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. TRANSFORMATIONS OF FUNCTIONS. By applying certain transformations

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FUNCTIONS AND MODELS

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FUNCTIONS AND MODELS

1.3

New Functions from Old Functions

  • In this section, we will learn:
  • How to obtain new functions from old functions
  • and how to combine pairs of functions.
slide3
TRANSFORMATIONS OF FUNCTIONS
  • By applying certain transformations
  • to the graph of a given function,
  • we can obtain the graphs of certain
  • related functions.
    • This will give us the ability to sketch the graphs of many functions quickly by hand.
    • It will also enable us to write equations for given graphs.
slide4
TRANSLATIONS
  • If c is a positive number,
  • then the graph of
  • y = f(x) + c is just the
  • graph of y = f(x) shifted
  • c units upward.
  • The graph of y = f(x - c) is
  • just the graph of y = f(x)
  • shifted c units to the right.
slide5
STRETCHING AND REFLECTING
  • If c > 1, then the graph
  • of y = cf(x) is the graph
  • of y = f(x) stretched by
  • a factor of c in the vertical
  • direction.
  • The graph of y = f(cx),
  • compress the graph of
  • y = f(x) horizontally by a
  • factor of c.
  • The graph of y = -f(x),
  • reflect the graph of
  • y = f(x) about the x-axis.

Figure 1.3.2, p. 38

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TRANSFORMATIONS
  • The figure illustrates these stretching
  • transformations when applied to the cosine
  • function with c = 2.

Figure 1.3.3, p. 38

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TRANSFORMATIONS
  • For instance, in order to get the graph of
  • y = 2 cos x, we multiply the y-coordinate of
  • each point on the graph of y = cos x by 2.

Figure 1.3.3, p. 38

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TRANSFORMATIONS

Example

  • This means that the graph of y = cos x
  • gets stretched vertically by a factor of 2.

Figure 1.3.3, p. 38

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TRANSFORMATIONS

Example 1

  • Given the graph of , use
  • transformations to graph:
  • a.
  • b.
  • c.
  • d.
  • e.

Figure 1.3.4a, p. 39

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Solution:

Example 1

  • The graph of the square root function

is shown in part (a).

  • by shifting 2 units downward.
  • by shifting 2 units to the right.
  • by reflecting about the x-axis.
  • by stretching vertically by a factor of 2.
  • by reflecting about the y-axis.

Figure 1.3.4, p. 39

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TRANSFORMATIONS

Example 2

  • Sketch the graph of the function
  • f(x) = x2 + 6x + 10.
    • Completing the square, we write the equation of the graph as: y = x2 + 6x + 10 = (x + 3)2 + 1.
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Solution:

Example 2

  • This means we obtain the desired graph by starting with the parabola y = x2 and shifting 3 units to the left and then 1 unit upward.

Figure 1.3.5, p. 39

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TRANSFORMATIONS

Example 3

  • Sketch the graphs of the following
  • functions.
  • a.
  • b.
slide14
Solution:

Example 3 a

  • We obtain the graph of y = sin 2x from that
  • of y = sin x by compressing horizontally by
  • a factor of 2.
    • Thus, whereas the period of y = sin x is 2 , the period of y = sin 2x is 2 /2 = .

Figure 1.3.6, p. 39

Figure 1.3.7, p. 39

slide15
Solution:

Example 3 b

  • To obtain the graph of y = 1 – sin x ,
  • we again start with y = sin x.
    • We reflect about the x-axis to get the graph ofy = -sin x.
    • Then, we shift 1 unit upward to get y = 1 – sin x.

Figure 1.3.7, p. 39

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TRANSFORMATIONS
  • Another transformation of some interest
  • is taking the absolute valueof a function.
    • If y = |f(x)|, then, according to the definition of absolute value, y = f(x) when f(x) ≥ 0 and y = -f(x) when f(x) < 0.
    • The part of the graph that lies above the x-axis remains the same.
    • The part that lies below the x-axis is reflected about the x-axis.
slide17
TRANSFORMATIONS

Example 5

  • Sketch the graph of the function
    • First, we graph the parabola y = x2 - 1 by shifting the parabola y = x2 downward 1 unit.
    • We see the graph lies below the x-axis when -1 < x < 1.

Figure 1.3.10a, p. 41

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Solution:

Example 5

  • So, we reflect that part of the graph about the x-axis to obtain the graph of y = |x2 - 1|.

Figure 1.3.10b, p. 41

Figure 1.3.10a, p. 41

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COMBINATIONS OF FUNCTIONS
  • Two functions f and g can be combined
  • to form new functions f + g, f - g, fg, and
  • in a manner similar to the way we add,
  • subtract, multiply, and divide real numbers.
slide20
SUM AND DIFFERENCE
  • The sum and difference functions are defined by:
  • (f + g)x = f(x) + g(x) (f – g)x = f(x) – g(x)
    • If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection .
    • This is because both f(x) and g(x) have to be defined.
slide21
SUM AND DIFFERENCE
  • For example, the domain of
  • is and the domain of
  • is .
  • So, the domain of
  • is .
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PRODUCT AND QUOTIENT
  • Similarly, the product and quotient
  • functions are defined by:
    • The domain of fg is .
    • However, we can’t divide by 0.
    • So, the domain of f/g is
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PRODUCT AND QUOTIENT
  • For instance, if f(x) = x2 and g(x) = x - 1,
  • then the domain of the rational function
  • is ,
  • or
slide24
COMBINATIONS
  • There is another way of combining two functions : composition - the new function is composed of the two given functions f and g.
    • For example, suppose that and
    • Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.
    • We compute this by substitution:
slide25
COMPOSITION

Definition

  • Given two functions f and g,
  • the composite function
  • (also called the composition of f and g)
  • is defined by:
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COMPOSITION
  • The domain of is the set of all x
  • in the domain of g such that g(x) is in
  • the domain of f.
    • In other words, is defined whenever both g(x) and f(g(x)) are defined.
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COMPOSITION
  • The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.
  • In other words, is defined whenever both g(x) and f(g(x)) are defined.

Figure 1.3.11, p. 41

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COMPOSITION

Example 6

  • If f(x) = x2 and g(x) = x - 3, find
  • the composite functions and .
    • We have:
slide29
COMPOSITION

Note

  • You can see from Example 6 that,
  • in general, .
    • Remember, the notation means that, first,the function g is applied and, then, f is applied.
    • In Example 6, is the function that firstsubtracts 3 and thensquares; is the function that firstsquares and thensubtracts 3.
slide30
COMPOSITION

Example 7

  • If and ,
  • find each function and its domain.
  • a.
  • b.
  • c.
  • d.
slide31
Solution:

Example 7 a

  • The domain of is:
slide32
Solution:

Example 7 b

  • For to be defined, we must have .
  • For to be defined, we must have ,that is, , or .
  • Thus, we have .
  • So, the domain of is the closed interval [0, 4].
slide33
Solution:

Example 7 c

  • The domain of is .
slide34
Solution:

Example 7 d

  • This expression is defined when both and .
  • The first inequality means .
  • The second is equivalent to , or , or .
  • Thus, , so the domain of is the closed interval [-2, 2].
slide35
COMPOSITION
  • It is possible to take the composition
  • of three or more functions.
    • For instance, the composite function is found by first applying h, then g, and then fas follows:
slide36
COMPOSITION

Example 8

  • Find if ,
  • and .
  • Solution:
slide37
COMPOSITION

Example 9

  • Given , find functions
  • f, g, and h such that .
    • Since F(x) = [cos(x + 9)]2, the formula for F states: First add 9, then take the cosine of the result, and finally square.
    • So, we let:
slide38
Solution:

Example 9

  • Then,
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