Compositions and Operations with Functions

Operations & Composition of Functions

Operations & Composition of Functions

​QUIZ REVIEW:

Composition of Functions
-using 2 or more functions to "build"
a new function

​

Multiple Choice

$$h\left(x\right)=3x+3$$  and  $$g\left(x\right)=-4x+1$$   $$Find\ \left(h+g\right)\left(0\right)$$  

• 2

• 1

• 4

• 3

Multiple Choice

$$If\ f\left(x\right)=1-x\ and\ g\left(x\right)=1+x$$   $$Find\ g\left(f\left(1\right)\right)$$  

• 1

• 4

• 3

• 2

Multiple Choice

$$If\ f\left(x\right)=x^2-1\ and\ g\left(x\right)=x^3$$   $$Find\ \left(f\ \circ\ g\right)\left(2\right)$$  

• 72

• 80

• 63

• 97

Multiple Choice

$$g\left(x\right)=x+2$$  and  $$f\left(x\right)=x^2-3x-10$$   $$Find\ \left(\frac{f}{g}\right)\left(x\right)$$  

• $$x+7$$

• $$x^2-2x-12$$  

• $$x+2$$  

• $$x-5$$  

Multiple Choice

$$If\ f\left(x\right)=\ \ \frac{1}{x}\ and\ h\left(x\right)=3x+2,$$  

$$find\ \left(f\circ h\right)\left(x\right).$$  

• $$f\left(h\left(x\right)\right)=3x-2$$  

• $$f\left(h\left(x\right)\right)=\frac{1}{3x+2}$$  

• $$f\left(h\left(x\right)\right)=3\left(\frac{1}{x}\right)+2$$  

• $$f\left(h\left(x\right)\right)=\frac{3}{x-2}$$  

Composition of Functions graphs

• EX: How to Find f(g(1)) = 6

• You will use BOTH graphs to arrive at the answer

• Work with inside function first, so go the "g" graph first and ACROSS to the input value x=1 and go UP to the find the output y-value (which is 3)

• That value 3 becomes the input for the outside function "f" so go ACROSS to input value x=3 on f and find the final output y-value (which is UP to 6)

Multiple Choice

$$g\left(f\left(-1\right)\right)$$  

• 1

• 3

• 2

• -5

Multiple Choice

$$If\ f\left(x\right)=1-x\ and\ g\left(x\right)=1+x$$   $$Find\ \left(g+f\right)\left(1\right)$$  

• 1

• 4

• 3

• 2

Multiple Choice

$$Find\ \left(g\ +\ f\right)\ \left(-1\right)$$  

• -1

• 3

• -2

• -5

Multiple Choice

$$f\left(g\left(-5\right)\right)$$  

• 1

• 3

• 2

• -5

Multiple Choice

$$Find\ \left(f\cdot g\right)\left(1\right)$$  

• 15

• 3

• -5

• -15

Composition of Functions table

Work Inside and then come outside.

The output of the inside becomes the input for the outside function

The answer to this problem =__

Operations with Functions

Like (f+g)(5) or (f*g)(5)

Simply substitute the numbers in...usually easier to do so FIRST and then add, subtract, multiply or divide the answers.

Multiple Choice

• A

• B

• C

• D

Multiple Choice

$$\left(h-g\right)\left(2\right)$$  

• 0

• 1

• 7

• 9

Multiple Choice

Find $$\left(f\cdot g\right)\left(3\right)$$  

• 88

• 0

• 66

• 154

Multiple Choice

Find $$-2\ f\left(x\right)$$  

• 18

• -4x² - 10x

• $$-6x^3$$  

• -4x² + 10x

Multiple Choice

Given g(x) = -2x+2 and f(x) = 3x²+4, find (g+f)(-3).

• 39
• -27
• -23
• -19

Multiple Choice

$$h\left(x\right)=x^2-3x-10$$  

$$j\left(x\right)=3x-2$$   What is  $$\left(\frac{h}{j}\right)\left(-2\right)$$  ?

• -2

• undefined

• 0

• 1

Multiple Choice

Given $$f\left(x\right)=5x+7$$  and  $$g\left(x\right)=3+4x$$ , find  $$\left(f\circ g\right)\left(x\right)$$  in simplest form.

• $$15+16x$$  

• $$20x^2+43x+21$$  

• $$22+20x$$  

• $$42+25x$$  

Multiple Choice

f(x) = 5x - 7

g(x) = 3x + 10

Find (f + g)(x).

• 8x² + 3

• 8x + 3

• 8x - 3

• 15x² - 70

Multiple Choice

f(x) = 5x - 7

g(x) = 3x + 10

Find (g - f)(x).

• 2x - 3

• -2x + 3

• -2x + 17

• 2x - 17

Multiple Choice

$$If\ f\left(x\right)=\ \ 2x\ +\ 1\ and\ h\left(x\right)=3x+2,$$  

$$find\ \left(f\circ h\right)\left(x\right).$$  

• $$=5x+3$$  

• $$=6x+5$$  

• $$=6x+2$$  

• $$=x+1$$  

Multiple Choice

f(x) = 3 and g(x) = -2x + 5 Find f(x) $$\cdot$$   g(x)

• 6x + 15

• 6x - 15

• -6x + 15

• 6x - 8

Multiple Choice

$$f\left(x\right)=x+2\ \ \ and\ \ \ \ g\left(x\right)=2x^2-2x$$  

$$Find\ \left(f-g\right)\left(x\right)$$  

• $$3x^3+2$$  

• $$2x^2+3x+2$$  

• $$2x^3+x^2+2x$$  

• $$-2x^2+3x+2$$