1.7 Combinations of Functions; Composite Functions

1. 1.7 Combinations of Functions;Composite Functions • Objectives • Find the domain of a function • Combine functions using algebra. • Form composite functions. • Determine domains for composite functions.

2. Find the indicated function values and determine whether the given values are in the domain of the function. f(1) and f(5), for f(1) = Since f(1) is defined, 1 is in the domain of f. f(5) = Since division by 0 is not defined, the number 5 is not in the domain of f.

3. Find the domain of the function Solution: We can substitute any real number in the numerator, but we must avoid inputs that make the denominator 0. Solve x2 3x  28 = 0. (x  7)(x + 4) = 0 x  7 = 0 or x + 4 = 0 x = 7 or x = 4 The domain consists of the set of all real numbers except 4 and 7 or {x|x  4 and x  7}.

4. To find the domain of a function that has a variable in the denominator, set the denominator equal to zero and solve the equation. All solutions to that equation are then removed from consideration for the domain.

5. Find the domain: Since the radical is defined only for values that are greater than or equal to zero, solve the inequality

6. Visualizing Domain and Range Keep the following in mind regarding the graph of a function: • Domain = the set of a function’s inputs, found on the x-axis (horizontal). • Range = the set of a function’s outputs, found on the y-axis (vertical).

7. Example Graph the function. Then estimate the domain and range. Domain = [1, ) Range = [0, )

8. The domain of a function is normally all real numbers but there are some exceptions: • A) You can not divide by zero. • Any values that would result in a zero denominator are NOT allowed, therefore the domain of the function (possible x values) would be limited. B) You can not take the square root (or any even root) of a negative number. Any values that would result in negatives under an even radical (such as square roots) result in a domain restriction.

9. Example • Find the domain • There are x’s under an even radical AND x’s in the denominator, so we must consider both of these as possible limitations to our domain.

14. Example • Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x) b) (f + g)(5) Solution: a) b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or (f + g)(5) = 3(5) + 7 = 22

15. Example • Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f - g)(x) b) (f - g)(5) Solution: a) b) We can find (f - g)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15 (f - g)(5) = f(5) - g(5) = 7 - 15 = -8 or (f - g)(5) = -(5) - 3 = -8

16. Example • Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (fg)(x) b) (fg)(5) Solution: a) b) We can find (fg)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15 (fg)(5) = f(5)g(5) = 7 (15) = 105 or (fg)(5) = 2(25) + 9(5) + 10 = 105

17. Given the functions below, find and give the domain. The radicand x – 3 cannot be negative. Solving gives

18. Composition of functions • Composition of functions means the output from the inner function becomes the input of the outer function. • f(g(3)) means you evaluate function g at x=3, then plug that value into function f in place of the x. • Notation for composition:

19. f o g f o g Given two functions f and g , the composite function , denoted by (read as “ f composed with g ”), is defined by The domain of is the set of x all numbers in the domain of g ( g such that x ) is in the domain of f .

20. 1 g ( x ) = f ( x ) = x x + 2 f o g Suppose and . Find .

21. 1 g ( x ) = f ( x ) = x x + 2 f o g . Find Suppose and the domain of .

22. Suppose that andfind

23. Suppose that andfind