Test Review Chapter 5 (Function Operations and Radical Functions)

Function Operations

100

What is the sum of f(x)=2x+3 and g(x)=x-5

= 3x-2

200

If f(x)=x² and g(x)=3x, what is (f*g)(x)?

= 3x³

300

If g(x)=x-4 and f(x)=3x-4, find f(2)/g(2)

=-1

400

If f(x)=x²-2x+5 and g(x)=5-4x, find (f-g)(x)

=x²+2x

500

What is the difference between (f-g)(x) and f(x)-g(x)?

They are the same, just slightly different notations

Inverse Functions

100

What is the inverse of f(x)=3x+2, 

= (x-2)/3 

= 1/3x-2/3

200

If f(x)=(x-1)³, what is the inverse

=cbrt(x)+1

300

What does it mean to find the inverse of a function algebraically? 

Answers can vary

400

What are the steps to determine if two functions are inverses of each other?

Find f(g(x)) and g(f(x)) to make sure that they are both equal to x. They both have to be checked since composition is not commutative. 

500

If f(x)=3sqrt(x-1), what is f^-1(x)?

(Hint there will need to be a domain restriction)

= (x/3)²+1 where x> 0

Composite Functions

100

Given f(x)=3x-4 and g(x)=x² , find g(f(1))

= 1

200

Given f(x)=x+1 and g(x)=2x, find (f o g)(3)

= 7

300

Given f(x)=2x+1 and g(x)=3x²-4x+3, find f(g(x))

= 6x²-8x+7

400

Define compostie functions

Where a function becomes the input of another function. 

Where the range of the substituted function becomes the domain for the other function. 

500

Compare and Contrast the following 

a) f(g(x))

b) (g o f)(x)

c) (f*g)(x)

Both a and b are composition using different notations where c is multiplication. But a and b are not the same since composition is not commutative and they switch the order. 

Graphing Radical Functions

100

What is the domain and range of the parent graph for all square root functions?

Domain: x > 0

Range: y > 0

200

Sketch the graph of f(x)=-sqrt(x-3), then state the domain and range

Graph

Domain: x>3

Range: y<0

300

What are the transformations that apply to this function? 

f(x)=-2cbrt(1/4(x+5))-3

Double if correctly transform (1,1)

Reflect x axis (Negative y's)

Vertical Stretch 2 (y *2)

Horizontal Stretch 4 (x*4)

Left 5

Down 3

(1,1) ->(1, -1)->(1, -2) -> (4, -2)->(-1, -5)

400

Sketch the graph of g(x)=sqrt(-3x)+4, then state the domain and range

Graph

Domain: x<0

Range: y>4

500

Sketch the graph of f^-1(x), given f(x)=2x²+4 where x> 0

Compare and contrast the transformations between f(x) and f^-1(x)

Graph

f(x) - Vertical Stretch 2, Up 4

f^-1(x) - Horizontal Stretch 2, Right 4

The amounts stay the same but any vertical transformation became a horizontal transformation

Solving Radical Equations

100

Solve and check your solution

sqrt(5x+3)=sqrt(2x+9)

x=2

Check 

sqrt(5(2)+3)=sqrt(2(2)+9)

sqrt(13)=sqrt(13)

200

Solve and check your solution. 

3cbrt(5x-3)=-6

x=-1

Check 3(cbrt(5(-1)-3))=-6

          3(cbrt(-8))=-6

               3(-2)=6

300

Is it possible to solve radical equations graphically rather than algebraically? Explain your answer. 

[Hint: use this test case sqrt(3x-5)+2=4-sqrt(x-1)]

Yes you can check by graphing the two sides of the equation and seeing the point of intersection which is (2,3) thus x=2 is the solution to the equation. 

400

Solve algebraically and check your solution(s). 

sqrt(2x-5) +4 = x

x=7 (x=3 is extraneous)

Check 

x=3.       x=7

5=3.       7=7

500

Explain the process and reasoning for ALWAYS checking your solutions for radical equations. 

We always check for extraneous solutions because when you are squaring you can add a solution since (-3)² and (3)² are both 9 but they did not come from the same number. The process is to plug the number(s) back into the original equation to see if it forms a true statement at the end, if it is not a true statement then you have found the extraneous solution.