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FUNCTIONS AND MODELS

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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  1. 1 FUNCTIONS AND MODELS

  2. 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.

  3. 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.

  4. 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.

  5. 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

  6. TRANSFORMATIONS • The figure illustrates these stretching • transformations when applied to the cosine • function with c = 2. Figure 1.3.3, p. 38

  7. 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

  8. 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

  9. TRANSFORMATIONS Example 1 • Given the graph of , use • transformations to graph: • a. • b. • c. • d. • e. Figure 1.3.4a, p. 39

  10. 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

  11. 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.

  12. 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

  13. TRANSFORMATIONS Example 3 • Sketch the graphs of the following • functions. • a. • b.

  14. 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

  15. 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

  16. 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.

  17. 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

  18. 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

  19. 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.

  20. 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.

  21. SUM AND DIFFERENCE • For example, the domain of • is and the domain of • is . • So, the domain of • is .

  22. 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

  23. PRODUCT AND QUOTIENT • For instance, if f(x) = x2 and g(x) = x - 1, • then the domain of the rational function • is , • or

  24. 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:

  25. COMPOSITION Definition • Given two functions f and g, • the composite function • (also called the composition of f and g) • is defined by:

  26. 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.

  27. 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

  28. COMPOSITION Example 6 • If f(x) = x2 and g(x) = x - 3, find • the composite functions and . • We have:

  29. 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.

  30. COMPOSITION Example 7 • If and , • find each function and its domain. • a. • b. • c. • d.

  31. Solution: Example 7 a • The domain of is:

  32. 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].

  33. Solution: Example 7 c • The domain of is .

  34. 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].

  35. 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:

  36. COMPOSITION Example 8 • Find if , • and . • Solution:

  37. 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:

  38. Solution: Example 9 • Then,

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