Advanced Algebra, 10.1 Operations and COmposition

Advanced Algebra,

10.1 Operations and Composition

By Jeremy Adelmann

Multiple Choice

Simplify.

$$\left(x^2+1\right)+\left(3x+5\right)$$  

• $$x^2+3x+6$$  

• $$x^2-3x-4$$  

• $$3x^3+5x^2+3x+5$$  

• $$\frac{x^2+1}{3x+5}$$  

• $$x-\frac{5}{3}+\frac{25}{9x+15}$$  

Multiple Choice

Simplify.

$$\left(x^2+1\right)-\left(3x+5\right)$$  

• $$x^2+3x+6$$  

• $$x^2-3x-4$$  

• $$3x^3+5x^2+3x+5$$  

• $$\frac{x^2+1}{3x+5}$$  

• $$x-\frac{5}{3}+\frac{25}{9x+15}$$  

Multiple Choice

Simplify.

$$\left(x^2+1\right)\left(3x+5\right)$$  

• $$x^2+3x+6$$  

• $$x^2-3x-4$$  

• $$3x^3+5x^2+3x+5$$  

• $$\frac{x^2+1}{3x+5}$$  

• $$x-\frac{5}{3}+\frac{25}{9x+15}$$  

Multiple Select

Simplify.

$$\left(x^2+1\right)\div\left(3x+5\right)$$  

• $$x^2+3x+6$$  

• $$x^2-3x-4$$  

• $$3x^3+5x^2+3x+5$$  

• $$\frac{x^2+1}{3x+5}$$  

• $$x-\frac{5}{3}+\frac{25}{9x+15}$$  

(f+g)(x) = f(x) + g(x)

= (x² + 12) + (3x - 7)

= ​x² + 12 + 3x - 7

= x²​ + 3x + 12 - 7

​

(f+g)(x) = x² + 3x + 5​

Given f(x) = x² + 12 and

g(x) = 3x - 7, find

​

(f+g)(x) = f(x) + g(x)

​

This simply means to add the two functions and combine like terms.​

Addition:

​

Adding and Subtracting Functions

Please take notes on the above information.

(f-g)(x) = f(x) - g(x)

= (x² + 12) - (3x - 7)

= (x²​ + 12) +(-3x + 7)

= ​x² + 12 - 3x + 7

= x²​ - 3x + 12 + 7

​

(f+g)(x) = x² - 3x + 19​

Given f(x) = x² + 12 and

g(x) = 3x - 7, find

​

(f-g)(x) = f(x) - g(x)

​

For subtraction, change the signs of each term in the functions, Then combine like terms.​

Subtraction:

​

Adding and Subtracting Functions

Please take notes on the above information.

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$f\left(x\right)=-4x-1$$   and $$g\left(x\right)=x^2\ +\ x$$  , find $$\left(f+g\right)\left(x\right)$$  

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$h\left(a\right)=a^2-3$$   and $$g\left(a\right)=3a+3$$  , find $$\left(h-g\right)\left(a\right)$$  

(f⋅g)(x)= f(x) ⋅ g(x)

= (x² -3x - 40)(x + 5)

= x²​(x+5) - 3x(x+5) - 40(x+5)

= x³​ +5x² -3x² +15x -40x -200

= x³​ - 2x² - 25x - 200

​

(f⋅g)(x) = x³ - 2x² - 25x -200​

Given f(x) = x² - 3x - 40 and

g(x) = x + 5, find

​

(f⋅g)(x) = f(x) ⋅ g(x)

​

For multiplication, multiply each term in the first function by each term in the second function (F.O.I.L). Then combine like terms.​

Multiply:

​

Multiply and Divide Functions

Please take notes on the above information.

(f/g)(x)= f(x)/g(x)

= (x² - 3x - 40)/(x + 5)

= (x² - 3x - 40)÷(x + 5)

​-5| 1 -3 -40

_____-5___40_ ​

​ 1 -8 | 0

(f/g)(x) = x - ​8

Given f(x) = x² - 3x - 40 and

g(x) = x + 5, find

​

(f/g)(x) = f(x)/g(x)

​

For division, make a fraction with one of the functions over the other. You can use synthetic division to divide the numberator by the denomiator.

Divide:

​

Multiply and Divide Functions

Please take notes on the above information.

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$g\left(n\right)=n-1$$   and $$h\left(n\right)=n^3-4$$  , find $$\left(g\times h\right)\left(n\right)$$  .

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$g\left(x\right)=-2x^2-5x-3$$   and $$h\left(x\right)=x-3$$  , find $$\left(\frac{g}{h}\right)\left(x\right)$$ .

​Example 1

(f⋄g)(x) = f[g(x)]

= 3( ) - 12

= 3(10x) - 12

​ = 30x - 12

​​

(f⋄g)(x) = 30x - 12

Given f(x) = 3x - 12 and

g(x) = 10x find

​

(f⋄g)(x) = f[g(x)]

​

For the composition, start by writing the first function, with an open set of parenthesis instead of the variable. In the paraenthsis, write the other function.

Composition:

​

Composition of Functions

Please take notes on the above information.

​Example 2

(g⋄f)(x) = g[f(x)]

= 10( )

= 10(3x - 12)

= 30x - 120

​​

(g⋄f)(x) = 30x - 120

Given f(x) = 3x - 12 and

g(x) = 10x find

​

(g⋄f)(x) = g[f(x)]

​

This is the same as Example 1, but you start with second function and you plug in the first equation.

Composition:

​

Composition of Functions

Please take notes on the above information.

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$f\left(x\right)=4x\ -\ 2$$   and $$g\left(x\right)=3x+4$$  , find $$\left(f\circ g\right)\left(x\right)$$ .

Fill in the Blank

Perform the indicated operation. (Use the carrot to indicate an exponent. For example, $$x^2+8$$   would be written as x^2 + 8.)

If $$h\left(x\right)=x\ -\ 4$$   and $$g\left(x\right)=4x+3$$  , find $$\left(f\circ g\right)\left(x\right)$$ .