1. Essential Question: How do you find the value of f(g(5)) and g(f(3)) given: f(x) = 3x + 1 and g(x) = 2x - 5

2.  Operations with functions  You can add, subtract, multiply and divide functions. Addition(f + g)(x) = f(x) + g(x) Multiplication(f g)(x) = f(x) g(x) Subtraction(f - g)(x) = f(x) – g(x) Division

3.  Domains  The domain of a function is all real numbers that can be substituted (plugged in) to a function in order to get an answer  The domain for most functions are all real numbers, with two significant exceptions You can’t use a number that would make you divide by 0. You can’t use numbers that would get you the square root of a negative number. Only the first rule will apply for us in this chapter.  The domain for the (addition, subtraction, and multiplication) combined functions are all numbers that work for both individual functions.

4.  Example #1  If f(x) = 3x + 8 and g(x) = 2x – 12  Find f + g, f – g, and their domains (f + g)(x) = f(x) + g(x) = (3x + 8) + (2x – 12) = 5x – 4 (f – g)(x) = f(x) – g(x) = (3x + 8) – (2x – 12) = 3x + 8 – 2x + 12 = x + 20  Because the domain for both functions are all real numbers, the domains of f + g and f – g are also all real numbers

5.  Example #2 – Y OUR T URN  If f(x) = 5x 2 – 4x and g(x) = 5x + 1  Find f + g, f – g, and their domains (f + g)(x) = (f – g)(x) =  The domain is: 5x 2 + x + 1 5x 2 - 9x - 1 All real numbers

6.  Example #3  If f(x) = x 2 - 1 and g(x) = x + 1  Find f g,, and their domains (f g)(x) = f(x) g(x) = (x 2 - 1)(x + 1) = x 3 + x 2 – x – 1 Division on next slide  Because the domain for both functions are all real numbers, the domain of f g is all real numbers

7.  Example #3  If f(x) = x 2 - 1 and g(x) = x + 1  Find f g,, and their domains  For this domain, we have to look at the original functions (both all real numbers) as well as the combined.  In this case, we can’t use -1, as that would cause us to divide by 0. So we say the domain is all real numbers except -1 (also written as x ≠ -1)

8.  Example #2 – Y OUR T URN  If f(x) = 6x 2 + 7x - 5 and g(x) = 2x - 1  Find f g,, and their domains (f g)(x) = =  The domain is: 12x 3 + 8x 2 - 17x + 5 3x + 5 Multiplication: All real numbers Division: All real numbers except ½

9.  Assignment  Page 400 – 401  1 – 19, all

10. Essential Question: How do you find the value of f(g(5)) and g(f(3)) given: f(x) = 3x + 1 and g(x) = 2x - 5

11.  Composite Functions  A composite function is the result of taking the results from one function, and then applying them into another.  It uses the symbol o, as like (f o g). Do not confuse the symbol with either the letter “o” or with the symbol for multiplication “”  (f o g)(x) means f(g(x)), meaning apply the x to the inner-most function first, apply that answer to the next closest function.  Like subtraction, the order the functions are presented in matters.

12.  Example #1, Method 1  If f(x) = x - 2 and g(x) = x 2  Find (g o f)(-5) (g o f)(-5) = g(f(-5)) f(-5) = (-5) – 2 = -7 = g(-7) = (-7) 2 = 49  We apply inside out: (g o f)(-5)

13.  Example #1, Method 2  We can combine the functions as variables before plugging numbers in  If f(x) = x - 2 and g(x) = x 2  Find (g o f)(-5) (g o f)(x) = g(f(x)) = g(x – 2) = (x – 2) 2 (g o f)(-5) = ((-5) – 2) 2 = (-7) 2 = 49

14.  Example #1 – Y OUR T URN  If f(x) = x - 2 and g(x) = x 2  Find (f o g)(-5) and (f o g)() (f o g)(-5) = (f o g)(x) = x 2 - 2 23

15.  Example #2 – Y OUR T URN  If f(x) = x 3 and g(x) = x 2 + 7  Find (g o f)(2) and (f o g)(2) (g o f)(2) = (f o g)(2) = 71 1331

16.  Real World Connection, Consumer Issues  Suppose you are shopping in the store. The store is offering 20% off everything in the store. You also have a coupon worth $5 off any item. a) Use function to model discounting an item by 20% and to model applying the coupon. b) Use a composition of your two functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon. c) Use a composition of your two functions to model how much you would pay for an item if the clerk applies the coupon first and then the discount. d) How much more is any item if the clerk applies the coupon first?

17. a) 20% discount:f(x) = x – 0.2x = 0.8x $5 coupon:g(x) = x – 5 b) Discount first:(g o f)(x) = g(f(x)) = g(0.8x) = 0.8x – 5 c) Coupon first:(f o g)(x) = f(g(x)) = f(x - 5) = 0.8(x – 5) = 0.8x – 4 d) Difference: (f o g)(x) - (g o f)(x) = (0.8x – 4) – (0.8x – 5) = 0.8x – 4 – 0.8x + 5 = 1 Meaning: Any item would cost you $1 more if the clerk applied the coupon first

18.  Assignment  Page 400 - 401  20 – 43, all

19.  Extra Examples / Quiz Practice – Y OUR T URN  Let: o f(x) = 2x + 3 o g(x) = x 2 – x o h(x) = 3x – 1 o j(x) = 2x 2  Find the following: o g(x) + h(x) = o 4f(x) – 2h(x) = o (j o g)(3) = o f(1) + g(2) – h(3) j(4) = x 2 + 2x - 1 2x + 14 72 -249