Operations on Functions f(x) + g(x) f(x) ∙ g(x) f(x) ÷ g(x) f(x) - g(x) ƒ(g(x)) Operations on Functions Lesson 2.5

Operations on 𝒇(𝒙) Find 𝟒𝒇(𝒙) [𝒇 𝒙 ] =[𝟐𝒙+𝟑] 𝟒[𝒇 𝒙 ] =𝟒[𝟐𝒙+𝟑] Rewrite with brackets around entire 𝒇(𝒙). Perform operation on entire quantity. Simplify. Find 𝟒𝒇(𝒙) [𝒇 𝒙 ] =[𝟐𝒙+𝟑] 𝟒[𝒇 𝒙 ] =𝟒[𝟐𝒙+𝟑] Which variable is 4 being added to? What do I mean when I say “entire dependent variable”?

Practice Complete the following problem at your table. 𝑔 𝑥 = 1 2 𝑥−2 Find 6𝑔 𝑥 −8 ANSWER: 6𝑔 𝑥 −8=3𝑥−20

Independent Practice Complete problem set A independently.

𝑓 𝒙 =2𝒙+3 𝑓(𝟒𝒙) =2 𝟒𝒙 +3 Operations on 𝑥 Find 𝑓(𝟒𝒙) Rewrite with space instead of x. Substitute input into that space. Simplify. 𝑓 𝒙 =2𝒙+3 Find 𝑓(𝟒𝒙) 𝑓(𝟒𝒙) =2 𝟒𝒙 +3 Which variable is 4 being added to? What will we substitute in for parentheses?

Practice Complete the following problem at your table. 𝑔 𝑥 = 1 2 𝑥−2 Find 𝑔(6𝑥−8) ANSWER: 𝑔 6𝑥−8 =3𝑥−6

Independent Practice Complete problem set B independently.

Operations on multiple functions: Adding and Subtracting Find: Sometimes written: 𝟐𝒙+𝟓 − 𝒙 𝟐 −𝟑𝒙−𝟏 ( ) What is being subtracted from 2x+5? If they say g(x), say what is g(x)? How do I write that I am subtracting 𝑥 2 −3𝑥−1? R Remember to subtract entire quantity (distribute the negative)!

Operations on multiple functions: Multiplying Find (𝑓∙𝑔)(𝑥), fully simplified.

Practice Complete the following problems independently. 𝑔 𝑥 =2𝑥−4 and ℎ 𝑥 =−2𝑥+5. Find (ℎ−𝑔)(𝑥). 𝑔 𝑥 =2𝑥−4 and ℎ 𝑥 =−2𝑥+5. Find (𝑔∙ℎ)(𝑥). −𝟐𝒙+𝟗 −𝟒 𝒙 𝟐 +𝟏𝟖𝒙−𝟐𝟎

Independent Practice Complete problem set C independently.

Operations on Functions f(x) + g(x) f(x) ∙ g(x) f(x) ÷ g(x) f(x) - g(x) ƒ(g(x)) Operations on Functions Lesson 2.5b

DO NOW Review for the quiz today: Silently re-read and annotate your notes, HW assignments and classwork. Highlight key points and write down reminders for yourself.

Oral Drill Function or Not? {(6, -1), (-2, -3), (1,8), (-2,-5)} Not x Y a X b c d Z

Oral Drill Function or Not? Function

Oral Drill Domain and range of the following relations: {(6, -1), (-2, -3), (1,8), (-2,-5)} Domain: {6, -2, 1} Range: {-1, -3, 8, -5}

Oral Drill Domain and range of the following relations: Domain: {a, b, c, d} Range: {X, Y, Z} x Y a X b c d Z

Oral Drill Domain and range of the following relations: Domain: all real # Range: y ≤4

Oral Drill If f(x) = 3x+4, what is –f(x)? -f(x) = -3x – 4

Oral Drill Describe the transformations of h(x) = −5 − 1 3 𝑥+3 −2 -horizontal stretch by a factor of 1 3 -reflection about the y-axis -horizontal translation 3 units to the left -vertical stretch by a factor of 5 -reflection about the x-axis -vertical translation 2 units down

Quiz When you finish, organize your binder If you have extra time, please help organize a partner’s binder

Review 𝑓 𝑥 =3𝑥−5. 𝐹𝑖𝑛𝑑 −𝑓 𝑥 Is the input or output changing? Input – independent variable Put a space where the original input is! 𝑓 =3 −5 Substitute the new input. 𝑓 𝑥+1 =3 𝑥+1 −5 =3𝑥+3 −5 =3𝑥 −2

Review 𝑓 𝑥 =3𝑥−5. 𝐹𝑖𝑛𝑑 𝑓 𝑥+1 Is the input or output changing? Output – dependent variable Write the output, then operate! 𝑓 𝑥 =3𝑥−5 −𝑓 𝑥 =− 3𝑥−5 −𝑓 𝑥 =−3𝑥+5

Representing Operations Graphically Use the graph to find f(-2) + g(-2). Check your work by finding f(x) + g(x) algebraically. Then evaluate for x = -2  TIME PERMITTING 

Representing Operations Graphically 𝑓 𝑥 =𝑥−2 𝑔 𝑥 =−𝑥+3 Use the graph to find g(0) x f(0). g(0) x f(0) 3 × -2 -6 Check your work by finding g(x) x f(x) algebraically. Then evaluate for x=0 (𝑥−2)(−𝑥+3) −𝑥 2 +5𝑥−6 When x= 0: − 0 2 +5 0 −6 −6