# 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.

\(\sqrt3\)

\(\frac32\)

\(\sqrt[3]{2}\)

\(3^\frac12\)

\(3^2\)

### 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.

- Write an expression to represent the amount of medication, in milligrams, in the patient’s body at:
- 1 p.m. on the same day
- 7 p.m. on the same day

- 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:

- Each month?
- Every 3 months?
- 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.

- Express the car’s value, in dollars, as a function of time \(d\), in decades, since purchase.
- Write an expression to represent the car’s value 4 years after purchase.
- 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.

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

### Problem 6

Decide if each statement is true or false.

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

### 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?

$545.45

$3,748.78

$9,411.43

$1,124,634.54