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Rewrite in standard form, if possible.

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1 Rewrite in standard form, if possible.
Bellwork Rewrite in standard form, if possible. 1. (2x + 5) – (x2 + 3x – 2) 2. (x – 3)(x + 1)2 Evaluate. 3. 𝑓 π‘₯ =2 π‘₯ 2 βˆ’5; 𝑓𝑖𝑛𝑑 𝑓(βˆ’3) Find the domain of each function: 4. a) 𝑔 π‘₯ = 1 π‘₯βˆ’2 b) β„Ž π‘₯ = π‘₯βˆ’2 +5 5. Read 6.6

2 =βˆ’ 𝒙 𝟐 βˆ’π’™+πŸ• Combine like terms.
Bellwork Solutions Rewrite in standard form, if possible. 1. (2x + 5) – (x2 + 3x – 2) =2π‘₯+5βˆ’ π‘₯ 2 βˆ’3π‘₯+2 Distribute! =βˆ’ 𝒙 𝟐 βˆ’π’™+πŸ• Combine like terms.

3 Rewrite in standard form, if possible.
Bellwork Solutions Rewrite in standard form, if possible. 2. (x – 3)(x + 1)2 = π‘₯βˆ’3 π‘₯+1 π‘₯+1 Distribute! = π‘₯βˆ’3 ( π‘₯ 2 +π‘₯+π‘₯+1) Combine like terms. = π‘₯βˆ’3 ( π‘₯ 2 +2π‘₯+1) Distribute again! =π‘₯ π‘₯ 2 +2π‘₯+1 βˆ’3 ( π‘₯ 2 +2π‘₯+1) (Optional step) = π‘₯ 3 +2 π‘₯ 2 +π‘₯βˆ’3 π‘₯ 2 βˆ’6π‘₯βˆ’3 Combine like terms. = 𝒙 πŸ‘ βˆ’ 𝒙 𝟐 βˆ’πŸ“π’™βˆ’πŸ‘ Done.

4 Bellwork Solutions Evaluate. 3. 𝑓 π‘₯ =2 π‘₯ 2 βˆ’5; 𝑓𝑖𝑛𝑑 𝑓(βˆ’3)
3. 𝑓 π‘₯ =2 π‘₯ 2 βˆ’5; 𝑓𝑖𝑛𝑑 𝑓(βˆ’3) 𝑓 =2 ( ) 2 βˆ’5

5 Bellwork Solutions Evaluate. 3. 𝑓 π‘₯ =2 π‘₯ 2 βˆ’5; 𝑓𝑖𝑛𝑑 𝑓(βˆ’3)
3. 𝑓 π‘₯ =2 π‘₯ 2 βˆ’5; 𝑓𝑖𝑛𝑑 𝑓(βˆ’3) 𝑓 βˆ’3 =2 (βˆ’3) 2 βˆ’5 =2 9 βˆ’5 =18βˆ’5 =πŸπŸ‘

6 Bellwork Solutions Find the domain of each function: 4. a) 𝑔 π‘₯ = 1 π‘₯βˆ’2
Domain is the set of all possible inputs (x’s) for the function that will give real number outputs. In this case, π‘₯β‰ 2, because 𝑔 2 =𝑒𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑. So, the domain of 𝑔 π‘₯ 𝑖𝑠 𝐷:π‘₯β‰ 2 The domain of 𝑔(π‘₯) can also be written in interval notation as D: βˆ’βˆž,2 π‘ˆ(2,∞)

7 Bellwork Solutions Find the domain of each function: 4. a) 𝑔 π‘₯ = 1 π‘₯βˆ’2
So, the domain of 𝑔 π‘₯ 𝑖𝑠: π‘₯β‰ 2 The domain of 𝑔(π‘₯) can also be written in interval notation as D: βˆ’βˆž,2 π‘ˆ(2,∞). The π‘ˆ stands for union. For this problem, it would indicate that the possible inputs (x’s) would include everything from βˆ’βˆž up to, but not including, 2, and everything from 2 up to ∞.

8 Bellwork Solutions Find the domain of each function:
Domain is the set of all possible inputs (x’s) for the function that will give real number outputs. So, for this function, the smallest π‘₯ that will give a real number output is π‘₯=2. The domain of β„Ž(π‘₯) is 𝐷: π‘₯β‰₯2 The domain of β„Ž(π‘₯) can also be written in interval notation as D: [2, ∞)

9 Bellwork Solutions Find the domain of each function:
The domain of β„Ž(π‘₯) is 𝐷:π‘₯β‰₯2 The domain of β„Ž(π‘₯) can also be written in interval notation as D:[2, ∞) The [2 indicates that 2 is included in the domain of β„Ž(π‘₯).

10 Objectives Add, subtract, multiply, and divide functions.
Write and evaluate composite functions.

11 Vocabulary composition of functions

12 You can perform operations on functions in much the same way that you perform operations on numbers or expressions. You can add, subtract, multiply, or divide functions by operating on their rules.

13

14 Check It Out! Example 1a Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function. (f + g)(x) (f + g)(x) = f(x) + g(x) = (5x – 6) + (x2 – 5x + 6) Substitute function rules. = x2 Combine like terms.

15 Check It Out! Example 1b Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function. (f – g)(x) (f – g)(x) = f(x) – g(x) = (5x – 6) – (x2 – 5x + 6) Substitute function rules. = 5x – 6 – x2 + 5x – 6 Distributive Property = –x2 + 10x – 12 Combine like terms.

16 When you divide functions, be sure to note any domain restrictions that may arise.

17 Check It Out! Example 2a Given f(x) = x + 2 and g(x) = x2 – 4, find each function and its domain. (fg)(x) (fg)(x) = f(x) ● g(x) = (x + 2)(x2 – 4) Substitute function rules. = x3 + 2x2 – 4x – 8 Multiply. D: All real numbers or ℝ or (βˆ’βˆž,∞)

18 Given f(x) = x + 2 and g(x) = x2 – 4, find each function and its domain. Simplify if possible.
( ) (x) g f ( ) (x) g f g(x) f(x) = = x2 – 4 x + 2 Set up the division as a rational expression. = (x – 2)(x + 2) x + 2 Factor completely. Note that x β‰  –2. = (x – 2)(x + 2) (x + 2) Divide out common factors. = x – 2, where D: x β‰  –2 Simplify.

19 Another function operation uses the output from one function as the input for a second function. This operation is called the composition of functions.

20 The order of function operations is the same as the order of operations for numbers and expressions. To find f(g(3)), evaluate g(3) first and then substitute the result into f.

21 The composition (f o g)(x) or f(g(x)) is read β€œf of g of x.”
Anytime you see (f o g)(x) , immediately rewrite it is f(g(x)) . Reading Math

22 Be careful not to confuse the notation for multiplication of functions with composition fg(x) β‰  f(g(x)) Caution!

23 (f o g)(3) = f(g(3)) =𝑓 9 =2 9 βˆ’3 𝑔 3 =( 3) 2 =9 =18βˆ’3 =πŸπŸ“
Check It Out! Example 3a Given f(x) = 2x – 3 and g(x) = x2, find each value. (f o g)(3) = f(g(3)) =𝑓 9 =2 9 βˆ’3 =18βˆ’3 =πŸπŸ“ Immediately rewrite! When evaluating composite functions, we generally need to start on the inside and work our way out. In this case, start with finding 𝑔 3 : 𝑔 3 =( 3) 2 =9 Now, we can use 𝑔 3 to help us evaluate f(g(3)).

24 g(f(3)) =𝑔 3 = (3) 2 =πŸ— 𝑓 3 =2 3 βˆ’3=6βˆ’3=3 Check It Out! Example 3a
Given f(x) = 2x – 3 and g(x) = x2, find each value. g(f(3)) =𝑔 3 = (3) 2 =πŸ— When evaluating composite functions, we generally need to start on the inside and work our way out. In this case, start with finding 𝑓 3 : 𝑓 3 =2 3 βˆ’3=6βˆ’3=3 Now, we can use 𝑓 3 to help us evaluate g(f(3)).

25 You can use algebraic expressions as well as numbers as inputs into functions. To find a rule for f(g(x)), substitute the rule for g into f.

26 Check It Out! Example 4a Given f(x) = 3x – 4 and g(x) = , write each composite. State the domain of each. f(g(x)) f(g(x)) = 3( ) – 4 Substitute the rule g into f. = – 4 Distribute. Note that x β‰₯ 0. = Simplify. The domain of f(g(x)) is x β‰₯ 0 or [0,∞).

27 Substitute the rule f into g.
Check It Out! Example 4b Given f(x) = 3x – 4 and g(x) = , write each composite. State the domain of each. g(f(x)) g(f(x)) = Substitute the rule f into g. Note that x β‰₯ 4 3 = The domain of g(f(x)) is x β‰₯ or [ 4 3 ,∞). 4 3

28 Composite functions can be used to simplify a series of functions.

29 Check It Out! Example 5 During a sale, a music store is selling all drum kits for 20% off. Preferred customers also receive an additional 15% off. a. Write a composite function to represent the final cost of a kit for a preferred customer that originally cost c dollars. Step 1 Write a function for the final cost of a kit that originally cost c dollars. Drum kits are sold at 80% of their cost. f(c) = 0.80c

30 Check It Out! Example 5 Continued
Step 2 Write a function for the final cost if the customer is a preferred customer. g(c) = 0.85c Preferred customers receive 15% off.

31 Check It Out! Example 5 Continued
Step 3 Find the composition f(g(c)). f(g(c)) = 0.80(g(c)) Substitute g(c) for c. f(g(c)) = 0.80(0.85c) Replace g(c) with its rule. = 0.68c b. Find the cost of a drum kit at $248 that a preferred customer wants to buy. Evaluate the composite function for c = 248. f(g(c) ) = 0.68(248) The drum kit would cost $


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