Functions and models
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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

1

FUNCTIONS AND MODELS


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.


Functions and models

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.


Functions and models

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.


Functions and models

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


Functions and models

TRANSFORMATIONS

  • The figure illustrates these stretching

  • transformations when applied to the cosine

  • function with c = 2.

Figure 1.3.3, p. 38


Functions and models

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


Functions and models

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


Functions and models

TRANSFORMATIONS

Example 1

  • Given the graph of , use

  • transformations to graph:

  • a.

  • b.

  • c.

  • d.

  • e.

Figure 1.3.4a, p. 39


Functions and models

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


Functions and models

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.


Functions and models

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


Functions and models

TRANSFORMATIONS

Example 3

  • Sketch the graphs of the following

  • functions.

  • a.

  • b.


Functions and models

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


Functions and models

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


Functions and models

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.


Functions and models

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


Functions and models

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


Functions and models

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.


Functions and models

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.


Functions and models

SUM AND DIFFERENCE

  • For example, the domain of

  • is and the domain of

  • is .

  • So, the domain of

  • is .


Functions and models

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


Functions and models

PRODUCT AND QUOTIENT

  • For instance, if f(x) = x2 and g(x) = x - 1,

  • then the domain of the rational function

  • is ,

  • or


Functions and models

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:


Functions and models

COMPOSITION

Definition

  • Given two functions f and g,

  • the composite function

  • (also called the composition of f and g)

  • is defined by:


Functions and models

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.


Functions and models

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


Functions and models

COMPOSITION

Example 6

  • If f(x) = x2 and g(x) = x - 3, find

  • the composite functions and .

    • We have:


Functions and models

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.


Functions and models

COMPOSITION

Example 7

  • If and ,

  • find each function and its domain.

  • a.

  • b.

  • c.

  • d.


Functions and models

Solution:

Example 7 a

  • The domain of is:


Functions and models

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].


Functions and models

Solution:

Example 7 c

  • The domain of is .


Functions and models

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].


Functions and models

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:


Functions and models

COMPOSITION

Example 8

  • Find if ,

  • and .

  • Solution:


Functions and models

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:


Functions and models

Solution:

Example 9

  • Then,


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