Using Function Language and Notation

5.8.3 Using Function Language and Notation

5.8.3 • Using Function Language and Notation

Activity

An exponential function is written in the form $f (x) = a b^{x}$ , where $a$ is the initial value and $b$ is the growth factor.

1. In a biology lab, a population of 50 bacteria reproduce by splitting. Every hour, on the hour, each bacterium splits into two bacteria.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The number of bacteria is a function of time passed in hours.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of hours. Dependent: number of bacteria.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 50 \cdot 2^{t}$

2. Every year after a new car is purchased, it loses $\frac{1}{3}$ of its value. Let’s say that the new car costs $18,000.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The value of a car is a function of the number of years since purchase.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of years since purchase of car. Dependent: value of car in dollars.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 18, 000 \cdot (\frac{2}{3})^{t}$ . Note: $\frac{1}{3}$ is the rate, and growth factor $= 1 -$ rate, so the growth factor is $\frac{2}{3}$ because there’s $\frac{2}{3}$ left.

3. To control an algae bloom in a lake, scientists introduce some treatment products. The day they begin treatment, the area covered by algae is 240 square yards. Each day since the treatment began, a third of the previous day’s area (in square yards) remains covered by algae. Time $t$ is measured in days.

a. Write a sentence of the form “_____ is a function of _____.”

Compare your answer:

The area covered by algae is a function of the number of days since the treatment.

b. Indicate which is the independent and which is the dependent variable.

Compare your answer:

Independent: number of days since treatment. Dependent: area of algae in square yards.

c. Write an equation that represents the situation using function notation.

Compare your answer:

$f (t) = 240 \cdot (\frac{1}{3})^{t}$ . Note: $\frac{1}{3}$ is given as the growth factor because it represents what remains.

Video: Writing an Exponential Function from a Situation

Watch the following video to learn more about exponential functions:

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Self Check

A new computer is purchased for $2,500. Each year after its purchase, it loses

\frac15

of its value. Which equation represents this situation?

$f(t)=\frac{4}{5} \cdot (2500)^t$
$f(t)=2500 \cdot (\frac{4}{5})^t$
$f(t)=2500 \cdot (\frac{4}{5})t$
$f(t)=2500+(\frac{4}{5})^t$

Additional Resources

Writing Functions from Exponential Situations

EXPONENTIAL GROWTH

A function that models exponential growth grows by a rate proportional to the amount present. For any real number $x$ and any positive real numbers $a$ and $b$ such that $b \neq 1$ , an exponential growth function has the form

$f (x) = a b^{x}$

where

$a$ is the initial or starting value of the function.
$b$ is the growth factor or growth multiplier per unit $x$ .

Example 1

The starting value of the function is 10. This function doubles every year.

Growth factor function	$𝑓 (𝑥) = 𝑎 b^{x}$
Identify the initial value.	$a = 10$
Identify the growth factor.	$b = 2$
Identify the unit.	$x = t$ for time in years.
Write the exponential equation.	$𝑓 (y) = 10 \cdot 2^{t}$

Steps to define an exponential growth function by identifying its components: initial value, growth factor, and unit.

Example 2

An acre of land is worth $40,000 at the time of purchase. It doubles in value every decade. Write an equation for this situation where $t$ is time in decades.

Identify the initial value.	$a = 40, 000$
Identify the growth factor.	$b = 2$
Write the exponential equation.	$f (t) = 40, 000 \cdot 2^{t}$

This table illustrates the steps to identify the initial value, growth factor, and formulate the corresponding exponential equation.

Try it

Writing Functions from Exponential Situations

1. The starting value of the function is 150. This function triples every decade

Compare your answer:

Here is how to write an exponential function from a situation:

Identify the initial value.	$a = 150$
Identify the growth factor.	$b = 3$
Write the exponential equation.	$f (y) = 150 \cdot 3^{t}$

Illustrates the process of identifying key parameters and constructing an exponential equation.

2. When first measured, the population of deer in a forest is 40. Over time, the population triples every year. Write an exponential equation where $t$ is time in years.

Compare your answer:

Here is how to write an exponential function from a situation:

Identify the initial value.	$a = 40$
Identify the growth factor.	$b = 3$
Write the exponential equation.	$f (t) = 40 \cdot 3^{t}$

This table demonstrates how to identify key parameters like initial value and growth factor to construct an exponential equation.