1. WARM UP Let f(x) = write the equation for g(x) in terms of x. 1.g(x) = -3 + f(x) 2. g(x) = f(1/3x) 3.g(x) = ½f(x + 3)

2. COMPOSITION OF FUNCTIONS

3. OBJECTIVES Given two functions, graph and evaluate the composition of one function and then the other. Evaluate a composite function at various values and then find an explicit formula for f(g(x)). Find the domain and range of a composite function.

4. TERMS & CONCEPTS Composite function Input Output Inside function Outside function Notation for a composite function:, f(g(x)), f ° g(x), (f ° g)(x) Domain & range of a composite function

5. COMPOSITE FUNCTION If you drop a pebble into a pond, a circular ripple extends out from the drop point (Figure 1-4a). The radius of the circular ripple is a function time. The area enclosed by the circular ripple is a function of the radius. So area is a function of time through this chain of functions: Area depends on radius. Radius depends on time. Figure 1-4a In this case, the area is a composite function of time. You will now learn some of the mathematics of composite functions.

6. SYMBOLS OF COMPOSITE FUNCTIONS Suppose that the radius of the ripple is increasing at a constant rate of 8 inches per second. Then where r is the number of inches and t is the number of seconds. If t = 5, then r = 8t The area of the circular region is given by a = πr r = 8  5 = 40 in. where a is the area in square inches and r is the radius. At time t = 5, when the radius is 40, the area is given by a = π  40 = 1600π = 5026.5482…≈ 5027 in. Or about 35 ft.5027/144 = 34.909722..

7. DEFINING COMPOSITION The 8 is the input for the radius function, and the 40 in. is the output. The figure shows that the output of the radius function becomes the input for the area function. The output of the area function is 5026.5… Mathematicians often use f(x) terminology for composite functions. For these radius and area functions, you can write r = 8t a = πr Input for radius fn. 5s Output from radius fn. 40 in. Input for area fn. Output from area fn. 5027 in. Radius function Area function r(x) = 8x x is the input for function r. a(x) = πr x is the input for function a. The r and a become the names of the functions, and the r(x) and a(x) are the values or outputs of the functions.

8. DEFINING COMPOSITION The x simply stands for the input of the function. You must keep in mind that the input for function r is the time and the input for function a is the radius. Combining the symbols leads to this way of writing a composite function The x is the input for the radius function and the r(x) is the input for the area function. This symbol for area is pronounced “a of r of x.” Function r is called the inside function because it appears inside a pair of parentheses. Function a is called the outside function. the figure below shows this symbol and its meanings. The names parallel the terms inside transformation and outside transformation that you learned in the previous chapter. Area = a(r(x)) input for function r output of function r input for function r

9. DEFINING COMPOSITION The two function names are sometimes combined this way: a r(x) or (a r)(x) The symbol a r is pronounced “a composition r”. The parenthesis in the expression (a r)(x) indicate that a r is the name of the function.

10. COMPOSITE FUNCTIONS FROM GRAPHS This example shows you how to find a value of the composite function f(g(x)) from graphs of the two functions f and g. Example 1: Functions f and g are graphed below. Find f(g(30)), showing on copies of the graphs how you found this value.

11. SOLUTION First find the value of the inside function, g(30). g(30) ≈ 2.8 Use this output of function g as the input for function f, as shown on the right First find the value of the inside function, g(30). g(30) ≈ 2.8 Use this output of function g as the input for function f, as shown on the right. Note that x in f(x) is simply the input for function f and is not the same number as the x in g(x).

12. SOLUTION f(2.8) ≈ 180 f(g(30)) ≈ 180 f(2.8) ≈ 180 g(30) ≈ 2.8 f(x) g(x)

13. COMPOSITE FUNCTIONS FROM TABLES Example 2 shows you how to find values of a composite function when the two functions are defined numerically. a)Find f(g(x)) for the six values of x in the table. b)Find g(f(2)) and show that it does not equal f(g(2)). xf(x)g(x) 135 243 362 421 507 614

14. SOLUTION a) To find f(g(1)), first find the value of the inside function, g(1) by finding 1 in the x-column and g(x) column (third column). g(1) = 5 xf(x)g(x) 135 243 362 421 507 614 Then use 5 as the input for the outside function f by finding 5 in the x-column and f(5) in the f(x) column (second column). F(5) = 0 F(g(1)) = 0 Find the other values the same way. Here is a compact way to arrange your work f(g(1))=f(5) = 0 f(g(2))=f(3) = 6 f(g(3))=f(2) = 4 f(g(4))=f(1) = 3 f(g(5))=f(7), which does not exist f(g(6))=f(4) = 2

15. SOLUTION b) g(f(2)) = g(4) = 1, which is not the same as f(g(2)) = 6 xf(x)g(x) 135 243 362 421 507 614 Note that in order to find a value of a composite function such as f(g(x)), the value of g(x) must be in the domain of the outside function, f. Because g(5) = 7 in Example 2 and there is no value for f(7), the value of f(g(5)) is undefined.

16. COMPOSITE FUNCTIONS FROM EQUATIONS Let f be the linear function f(x) = 3 x + 5, and let g be the exponential function g(x) = a.Find f(g(4)), f(g(0)), and f(g(-1)) b.Find g(f(-1)) and show that it is not the same as f(g(-1)) c.Find an equation for h(x) = f(g(x)) explicitly in terms of x. Show that h(4) agrees with the value you found for f(g(4)). Example 3 shows you how to find values of a composite function if you know the equations of the two functions

17. SOLUTION a), and f(16) = 3  16 + 5 = 53, so f(g(4)) = 53 Writing the same steps more compactly for the other two values of x gives: b) g(f(-1)) = g(3  -1 + 5) = g(2) = 2 = 4, which does not equal 6.5 from part a.

18. SOLUTION a)f(f(2)) = f(3  2 + 5) = f(11) = 3  11 + 5 = 38 b)f(g(f(-3)) = f(g(3  -3 + 5) = f(g(-4)) = f(2 ) = f(0.0625) = 3  0.0625 + 5 = 5.1875

19. SOLUTION c) The equation is h(x) =

20. EXAMPLE 4 Example 4 shows you that you can compose a function with itself or compose more than two functions. Let f be the linear function f(x) = 3x + 5, and let g be the exponential fnction g(x) = 2, as in Example 3. Find these values: a)f(f(2)) b)f(g(f(-3))

21. DOMAIN & RANGE OF COMPOSITE FUNCTION In Example 2 you saw that the value of the inside function sometimes is not in the domain of the outside function. This example shows you how to find the domain of a composite function and the corresponding range under this condition.

22. EXAMPLE 5 The left figure below shows function g with domain 2 < x < 7, and the right side shows function f with domain 1 < x < 5. a. Show on copies of these graphs what happens when you try to find f(g(6)), f(g(8)) and f(g(2)). b. Make a table of values of g(x) and f(g(x)) for integer values of x from 1 thru 8. if there is no value, write “none.” From the table, what does the domain of function f of g seem to be? c. Find the domain of f of g algebraically & show that it agrees with part c.

23. SOLUTIONS a. The left graph shows that g(2) = -1 and g(6) = 3 but g(8) does not exist because 8 is outside the domain of function g. The right graph shows the two output values of function g, -1, and 3, used as inputs for function f Summarize the results: f(g(6)) = f(3) = 2 f(g(8)) does not exist f(g(2)) = f(-1), which does not exist

24. SOLUTIONS b. The domain of f of g seems to be 4 < x < 7. xg(x)f(g(x)) 1Nonenone 2none 30 416 524 632 740 8Nonenone c. To calculate the domain algebraically, first observe that g(x) must be within the domain of f.

25. SOLUTIONS Write g(x) in the domain of f. 1 < g(x) < 5 c. To calculate the domain algebraically, first observe that g(x) must be within the domain of f. 1 < x - 3 < g(x) Substitute x – 3 for g(x). 4 < x < 8 Add 3 to all three member of the inequality

26. SOLUTION Next observe that x must also be in the domain of g, specifically 2 < x < 7. The domain of f of g is the intersection of these two intervals. Number- line graphs will help you visualize the intersection. the domain of f of g is 4 < x < 8.

27. DEFINITION & PROPERTIES Note: Horizontal dilations and translations are examples of composite functions because they are inside transformations applied to x. For instance, the horizontal translation g(x) = 3(x – 2) is actually a composite function with the inside function f(x) = x – 2. Composite Function: The composite function f g (pronounced “f composition g”) is the function (f g)(x) = f(g(x)) Function g, the inside function, is evaluated first using x as its input. Function f, the outside function, is evaluated next, using g(x) as its input (the output of function g. The domain of f g is the set of all values of x in the domain of g for which g(x) is in the domain of f. The figue shows this relationship.

28. CH. 1.4 ASSIGNMENTS Textbook pg. 29 #1, 2, 3 & 5