# Lesson 4

Representing Functions at Rational Inputs

• Let’s find how quantities are growing or decaying over fractional intervals of time.

### Problem 1

A bacteria population is tripling every hour. By what factor does the population change in $$\frac12$$ hour? Select all that apply.

A:

$$\sqrt3$$

B:

$$\frac32$$

C:

$$\sqrt[3]{2}$$

D:

$$3^\frac12$$

E:

$$3^2$$

### Problem 2

A medication has a half-life of 4 hours after it enters the bloodstream. A nurse administers a dose of 225 milligrams to a patient at noon.

1. Write an expression to represent the amount of medication, in milligrams, in the patient’s body at:
1. 1 p.m. on the same day
2. 7 p.m. on the same day
2. The expression $$225 \boldcdot \left(\frac12\right)^{\frac52}$$ represents the amount of medicine in the body some time after it is administered. What is that time?

### Problem 3

The number of employees in a company has been growing exponentially by 10% each year. By what factor does the number of employees change:

1. Each month?
2. Every 3 months?
3. Every 20 months?

### Problem 4

The value of a truck decreases exponentially since its purchase. The two points on the graph shows the truck’s initial value and its value a decade afterward.

1. Express the car’s value, in dollars, as a function of time $$d$$, in decades, since purchase.
2. Write an expression to represent the car’s value 4 years after purchase.
3. By what factor is the value of the car changing each year? Show your reasoning.

### Problem 5

The value of a stock increases by 8% each year.

1. Explain why the stock value does not increase by 80% each decade.
2. Does the value increase by more or less than 80% each decade?

### Problem 6

Decide if each statement is true or false.

1. $$50^\frac12 = 25$$
2. $$\sqrt{30}$$ is a solution to $$y^2 = 30$$.
3. $$243^{\frac13}$$ is equivalent to $$\sqrt[3]{243}$$.
4. $$\sqrt{20}$$ is a solution to $$m^4 = 20$$.
(From Unit 4, Lesson 3.)

### Problem 7

Lin is saving $300 per year in an account that pays 4.5% interest per year, compounded annually. About how much money will she have 20 years after she started? A:$545.45

B:

$3,748.78 C:$9,411.43

D:

\$1,124,634.54

(From Unit 2, Lesson 26.)