6 Pull It All Together

BIG Idea Solving Equations and Inequalities

Solving an equation is the process of rewriting the equation to make what it says about its variables as simple as possible.

Task 1

An environmental equipment supplier sells hemispherical holding ponds for treatment of chemical waste. The volume of a pond is V 1 = 1 2 ( 4 3 π r 1   3 ) , v sub 1 , equals , 1 half . open . 4 thirds , pi , r sub 1 , cubed . close . comma  where r 1 r sub 1 is the radius in feet. The supplier also sells cylindrical collecting tanks. A collecting tank fills completely and then drains completely to fill the empty pond. The volume of the tank is V 2 = 12 π r 2 2 , v sub 2 , equals . 12 pi r sub 2 squared . comma  where r 2 r sub 2  is the radius of the tank.

1. Since V 1 = V 2 , v sub 1 , equals , v sub 2 , comma  write an equation that shows r 1 r sub 1  as a function of r 2 . r sub 2 , .  Write an equation that shows r 2 r sub 2  as a function of r 1 . r sub 1 , .
2. You want to double the radius of the pond. How will the radius of the tank change?

BIG Idea Solving Equations and Inequalities

The numbers and types of solutions vary based on the type of equation.

BIG Idea Function

You can represent functions in a variety of ways (such as graphs, tables, equations, or words). Each representation is particularly useful in certain situations.

Task 2

Suppose f ( x ) = x + 1 . f , open x close , equals , square root of x plus 1 end root , .

1. What are the domain and range of f?
2. Find f − 1 ( x ) . f super negative 1 end super , open x close .  What are its domain and range? Be careful!
3. Show that ( f ∘ f − 1 ) ( a ) = a = ( f − 1 ∘ f ) ( a ) open f composition , f super negative 1 end super , close open eh close equals eh equals open , f super negative 1 end super , composition f close open eh close  for any a in the respective domains.
4. Solve the equation f ( x ) = f − 1 ( x ) . f open x close equals , f super negative 1 end super , open x close .  Remember to check for extraneous roots.
5. Graph the functions f and f − 1 . f super negative 1 end super , .  Be sure that you accurately represent the domains of each function. Interpret graphically the solution(s) you found to the equation in part (d).

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