Learn – Algebra/Precalculus – Exponentials

1 Linear functions
- 1.1 Real-world scenarios
  - 1.1.1 Exercises
    - 1.1.1.1 Problem set
    - 1.1.1.2 Problem set
- 1.2 Using characteristic statement
  - 1.2.1 Exercises
2 Exponential functions

Linear functions

Real-world scenarios

Exercises

Problem set

In each of the following real-world scenarios, identify the independent quantity and the dependent quantity. Then, give the linear function’s characteristic statement using the real-world quantities. Then, analyze the statement based on your real-world knowledge to see if the function is linear.

Money earned in a summer job depending on the number of hours worked
Amount of money a car salesman makes depending on the number of cars sold
Number of pages a printer prints depending on the number of minutes it prints
Amount of food consumed by a person depending on his age
Time taken to drive depending on the distance to drive
Rent paid for a house depending on the number of rooms of house
Pounds of fertilizer needed depending on the area of backyard
Height of snow accumulation depending on the number of hours it snowed

Problem set

Emily opens a bank account with a deposit of $\$30000$ . Her plan is for her monthly income of $\$3000$ to be deposited into her bank account, and for her not to do any withdrawals. Express the bank account balance she should expect as a function of months that elapse from the time of opening the account.
Miller currently had a bank balance of $\$150000$ in the year 2015. His yearly income of $\$95000$ gets deposited into his bank account. He also withdraws $\$2500$ every month from his account for his monthly expenses. Model Miller’s bank account balance as a function of the number of years that elapsed since 2015.
Steve has a current bank account balance of . He plans to use this bank account to save the money that he makes from a side-job. He currently makes per year from the side job, and he is expected to get an yearly raise of on this job.
1. Model Steve’s income from the side-job as a function of number of years that elapse from now.
2. Model Steve’s bank account balance as a function of number of years that elapse from now.

Using characteristic statement

Exercises

Problem set

Given that $f(x) = 7x+I$ , in each of the following problems, express $?$ in terms of $A$ .

If $f(2010) = A$ , $f(2011) = ?$ .
If $f(511) = A$ , $f(512) = ?$ .
If $f(-10) = A$ , $f(-9) = ?$ .
If $f(100) = A$ , $f(99) = ?$ .
If $f(729) = A$ , $f(728) = ?$ .

Problem set

Given that $g(x) = 9x+I$ , find out what comes in place of $?$ in each of the following problems.

$9+g(21) = g(?)$
$81+g(100) = g(?)$
$9+g(?) = g(1000)$
$27+g(?) = g(181)$
$g(50)-9 = g(?)$
$g(1234)-45 = g(?)$
$g(?)-18 = g(91)$

Problem set

Given that $g(x) = 2x+5$ , find out what comes in place of $?$ in each of the following problems.

If $g(2010) = A$ , $g(2019) = ?$ .
$g(400) = ? + g(200)$
If $g(a) = B$ , $g(a+n) = ?$ .

Problem set

If $f(x) = 300 - 3x$ and $f(100) = A$ , what is $f(300)$ ?
If $f(x) = 100 - 10x$ and $f(10) = A$ , what is $f(100)$ ?
If $f(x) = 300 + 3x$ and $f(300) = A$ , what is $f(100)$ ?
If $f(x) = 100 + 10x$ and $f(10) = p + f(100)$ , what is the value of $p$ ?

Problem set

For each of the following problems, assume $f(x)$ is a linear function, and find the required value in terms of $A$ .

If $f(10)=21, f(11)=27, f(123) = A$ , find $f(127)$ .
If $f(123)=31, f(124)=41, f(-123) = A$ , find $f(-127)$ .
If $f(100)=100, f(200)=120, f(1003) = A$ , find $f(1004)$ .
If $f(150)=100, f(250)=150, f(0) = A$ , find $f(4)$ .
If $f(-100)=100, f(0)=300, f(1003) = A$ , find $f(993)$ .
If $f(100)=300, f(50)=100, f(441) = A$ , find $f(466)$ .
If $f(100)=300, f(200)=100, f(2003) = A$ , find $f(3003)$ .
If $f(917)=653, f(897)=733, f(1000) = A$ , find $f(1018)$ .

Problem set

For each of the following problems, assume $f(x)$ is a linear function.

If $f(7)=25$ and $f(10)=34$ , find the expression for $f(x)$ .
If $f(100)=55$ and $f(200)=0$ , find the expression for $f(x)$ .
If $f(0) = I, f(11)=100, f(23)=100+A$ , find the expression for $f(x)$ in terms of $A$ and $I$ .

Exponential functions

Real-world scenarios

Exercises

Problem set

The bacteria population $p$ in a lab dish after time $t$ , given in minutes, is modeled by $p(t) = 2^t$ . Explain how the bacteria population changes over time.
The population of a town was recorded each year starting from the year 1900. The population $p$ , in millions of people, approximately followed the model $p(y) = A\times 1.1^y$ , where $y$ is the number of years that elapsed since 1900. What does $A$ signify in this model? Then, explain how the population changed over the years starting from 1900.
The bird population in a sanctuary as a function of number of years that elapsed since the introduction of birds into the sanctuary was modeled by the expression $f(x) = 50\times 1.08^x$ . Assuming the model is correct, explain how the bird population changed over time.
A radio-active substance’s mass $m$ over time $t$ , given in hours, is modeled by the function $m(t) = 1000\times \left(\frac{1}{2}\right)^t$ . Explain in layman’s language how the substance’s mass changes over time.
A radio-active substance’s mass $f(x)$ over time $x$ , given in years, is modeled by the function $f(x) = 500\times \left(\frac{70}{100}\right)^x$ . Explain in layman’s language how the substance’s mass changes over time.

Problem set

The population of a town in the year 2015 was $250000$ . Every year, it gets multiplied by a factor of $1.15$ . Model the population of the town as a function of the number of years that elapsed since 2015.
The population of a town in the year 2015 was $250000$ . Every year, it increases by $10\%$ . Model the population of the town as a function of the number of years that elapsed since 2015.
In 2022, the number of flu infections on Jan 1^st on an island was $100$ . From then on, the infections increased by $20\%$ every week. Give an expression for the number of infections $f(x)$ after $x$ weeks.

Problem set

The rabbit population $p$ in an experiment after time $t$ , given in days, is modeled by $p(t) = 20\times 2^{t/9}$ . Explain how the rabbit population changes over time.
A radio-active substance’s mass $m$ over time $t$ , given in hours, is modeled by the function $m(t) = M\times \left(\frac{1}{2}\right)^{t/H}$ . What do $M$ and $H$ indicate in this model?
A radio-active substance’s mass $m$ over time $t$ , given in hours, is modeled by the function $m(t) = 1000\times \left(\frac{1}{2}\right)^{t/3}$ . Explain how the mass of the radio-active substance changes over time.
The half-life of thorium-234 isotope is $24.1$ days. Assuming an initial mass of $10$ g, give an expression for the mass remaining of the substance as a function of number of days elapsed.
The half-life of iron-55 isotope is $2.756$ years. Assuming an initial mass of $75$ g, give an expression for the mass remaining of the substance as a function of number of years elapsed.
The population of a town grows by $10\%$ every $3$ years. If the population of the town was $1$ million in the year $2015$ , express the population of the town after $t$ years as a function of $t$ .

Problem set

A radio-active substance of mass $600$ g was installed in a lab. After a week, the substance was found to have decayed by $100$ g. Give an expression to model the mass of the substance as a function of time(measured in weeks) elapsed since the installation.
The mass of a radio-active substance goes from g to g in days.
1. Assuming an initial mass of $150$ g, give an expression for the mass remaining of the radio-active substance as a function of number of days elapsed.
2. Determine the half-life of the radio-active substance.
A radio-active substance of mass $9000$ g was set up in a lab to be monitored. It was observed to lose $30$ g in $30$ minutes. Give an expression to model the mass of the substance as a function of time(measured in minutes) elapsed since the set-up.
The mass of a radio-active substance goes down to in years.
1. Assuming an initial mass of $500$ g, give an expression for the mass remaining of the radio-active substance as a function of number of days elapsed.
2. Determine the half-life of the radio-active substance.
On November 3rd, a new virus was detected on an island with $16$ people testing positive. On November 10th, the same virus was found in $24$ people. Give an expression to model the number of infections as a function of the number of days that elapse from November 3rd.
The population of a town in 2015 was $120000$ . The population in 2020 was $150000$ . Give an expression to model the population of the town as a function of number of years that elapse from 2015.

Problem set

The balance in a type of bank account is modeled as $f(x) = P\times1.1^x$ , where $P$ is the amount deposited initially into the bank account and $x$ is the number of years elapsed since the initial deposit. The model assumes there are neither deposits nor withdrawals after the initial deposit. Explain the interest scheme that is being applied to this type of bank account.
The amount accumulated in a bank account $A$ in $t$ years when an initial amount of $P$ is deposited in the account is modeled by $A(t) = P\left(1+r\right)^{t}$ . Explain the model behavior. What does $r$ indicate in this model?
The balance in a type of bank account is modeled as $B(y) = P\times1.08^y$ , where $P$ is the amount deposited initially into the bank account and $y$ is the number of years elapsed since the initial deposit. The model assumes there are neither deposits nor withdrawals after the initial deposit. Explain the interest scheme that is being applied to this type of bank account.
Anthony deposits $\$5000$ into a Certificate of Deposit (CD), which guarantees a return of $\$ 5000\times 1.05^6$ after $6$ years. Explain the interest scheme that is being applied on the CD.
Jonathan deposits $\$6500$ into a Certificate of Deposit (CD), which guarantees a return of $\$ 6500\times 1.12^8$ after $8$ years. Explain the interest scheme that is being applied on the CD.
In one Unity bank, a Certificate of Deposit (CD) triples the deposit every $9$ years. Assuming an opening deposit of $\$1000$ , model the returns from the CD after $y$ years.

Problem set

A bank offers a Certificate of Deposit (CD). For each of the following cases, with being the principal that is put into the CD, model the CD balance as a function of number of years :
1. the CD compounds yearly with an interest rate of $6\%$ per year
2. the CD compounds half-yearly with an interest rate of $6\%$ per year
3. the CD compounds monthly with an interest rate of $6\%$ per year
In the above problem, which of the three cases accumulates most interest after $10$ years?
The amount accumulated in a bank account $A$ in $t$ years when an initial amount of $P$ is deposited in the account is modeled by $A(t) = P\left(1+\frac{r}{n}\right)^{nt}$ . What do $r$ and $n$ indicate in this model?
Elizabeth invests $\$10000$ into a financial instrument, and observes that the instrument after $t$ years returns the amount (in dollars) $10000\times 1.03^{2t}$ . What compounding period and what annual interest rate are being applied to the financial instrument?
Donald deposits $\$100000$ into his bank account, and observes that his account balance after $t$ years is given by the function $100000\times 1.001^{365t}$ . What compounding period and what annual interest rate is the bank employing for Donald’s bank account?
Deborah is setting up a bank account, and needs to choose between two types of bank accounts, both of which give an annual interest rate of $12\%$ . One of them compounds yearly, while the other compounds monthly. Which account must she choose if she wishes to grow her money more?
Delta bank’s Certificate of Deposit (CD) compounds monthly with an interest rate of $8\%$ per year, while Omega bank’s CD compounds quarterly with an interest rate of $8\%$ per year. Assuming $P$ is the principal that is put into each of the CD’s, model the balance from each of the CD’s as a function of number of years $t$ elapsed. In addition, answer which bank’s CD would yield a better return after $5$ years?
Narender opens an account at a bank that compounds monthly. The compounding scheme is such that the dollar amount in his account should double every three years. What is the interest rate?
Mr.Buckingham manages a bank that uses a daily compounding scheme for its customer accounts. If he promises to his customers that the money doubles every five years, what interest rate should he use on the accounts?

Using characteristic statement

Exercises

Problem set

Given that $g(x) = I\times3^x$ , in each of the following problems, express $?$ in terms of $A$ .

If $g(2010) = A$ , $g(2011) = ?$ .
If $g(75) = A$ , $g(76) = ?$ .
If $g(-10) = A$ , $g(-9) = ?$ .
If $g(100) = A$ , $g(99) = ?$ .
If $g(127) = A$ , $g(126) = ?$ .

Problem set

Given that $f(x) = I\times5^x$ , find out what comes in place of $?$ in each of the following problems.

$5f(2010) = f(?)$
$5^5\times f(123) = f(?)$
$5f(?) = f(200)$
$5^{100}\times f(?) = f(200)$
$\frac{f(100)}{5} = f(?)$
$\frac{f(100)}{5^5} = f(?)$
$\frac{f(?)}{5^{50}} = f(100)$

Problem set

Given that $f(x) = 2^x$ , find out what comes in place of $?$ in each of the following problems.

If $f(2010) = A$ , $f(2019) = ?$ .
$f(400) = ? \times f(200)$
If $f(a) = B$ , $f(a+n) = ?$ .

Problem set

If $f(x) = 1000\times 0.1^x$ and $f(90) = A$ , what is $f(100)$ ?
If $f(x) = 2\times\left(\frac{1}{2}\right)^x$ and $f(100) = A$ , what is $f(200)$ ?
If $f(x) = 3\times\left(\frac{1}{3}\right)^x$ and $f(300) = m\times f(100)$ , what is the value of $m$ ?
If $f(x) = 2000\times 2^x$ and $f(200) = A$ , what is $f(100)$ ?
If $f(x) = 300\times 3^x$ and $f(100) = a\times f(300)$ , what is the value of $a$ ?

Problem set

For each of the following problems, assume $g(x)$ is an exponential function, and find the required value in terms of $B$ .

If $g(10)=21, g(11)=63, g(99) = B$ , find $g(100)$ .
If $g(31)=100, g(32)=200, g(62) = B$ , find $g(64)$ .
If $g(29)=5, g(33)=80, g(100) = B$ , find $g(101)$ .
If $g(100)=2, g(104)=162, g(0) = I$ , find $g(3)$ .
If $g(107)=81, g(104)=3, g(1000) = B$ , find $g(1004)$ .
If $g(44)=375, g(41)=3, g(825) = B$ , find $g(823)$ .
If $g(90)=162, g(94)=2, g(900) = B$ , find $g(902)$ .
If $g(90)=800, g(93)=100, g(2000) = B$ , find $g(1996)$ .

Problem set

For each of the following problems, $g(x)$ is an exponential function.

If $g(0) = I, g(7)=5$ and $g(9)=125$ , find the expression for $g(x)$ in terms of $I$ .
If $g(3)=4$ and $g(6)=32$ , find the expression for $g(x)$ .
If $g(2)=250$ and $g(5)=2$ , find the expression for $g(x)$ .

Problem set

Given that $f(x) = 10\times 4^{x/6}$ , in each of the following problems, express $?$ in terms of $A$ .

If $f(2010) = A$ , $f(2016) = ?$
If $f(a) = A$ , $f(a+6) = ?$
If $f(1000) = A$ , $f(1024) = ?$
If $f(a) = A$ , $f(a+36) = ?$
If $f(2007) = A$ , $f(2019) = ?$

Problem set

Assuming $f(x) = 51\times 4^{x/6}$ , find the value of $?$ in each of the following.

$\frac{f(204)}{f(180)} = ?$
$f(x) = 10\times 2^{x/?}$
$f(x) = 10\times 16^{x/?}$

Graphing

Exercises

Problem set

Graph and identify domain and range for each of the following functions.

$2^x$
$2^x-2$
$-3^x$
$\frac{5^x}{2}$
$3^{x-1}$
$2\times 2^{x}+2$
$5\times 3^{x}-1$

Problem set

Graph and identify domain and range for each of the following functions.

$-0.5\times 3^{x}-1$
$-3\times 3^{x-2} - 3$
$2^{-x}$
$-5^{-x}$
$-2\times 3^{-x} - 4$
$e^x$
$2e^{-x} - 1$

Problem set

Graph and identify domain and range for each of the following functions.

$0.5^x$
$0.5^x+2$
$-2\times 0.5^x + 2$
$-0.5\times 0.5^{-x+2} - 5$

Continous-growth rate models

Exercises

Problem set

The bacteria growth in a dish is modeled as $P\times e^{kt}$ , where $t$ is time in hours and $P$ is the initial population of the bacteria. Explain the model.
Dr.Cheever is experimenting with bacteria in his lab. He starts with a population of $10000$ bacteria. He observes that the bacteria population follows the model $10000\times e^{0.5t}$ , where $t$ is time in days. Explain how the bacteria population is growing in Dr.Cheever’s experiment.
A radio-active substance’s mass is modeled by the function $M\times e^{-0.08t}$ , where $t$ is time in years. Explain the model.
In 2018, the number of flu infections on Novemeber 1^st on an island nation was . From then on, the infections had a continuous growth rate per week.
1. What were the number of infections $w$ weeks after November 1^st?
2. What were the number of infections $d$ days after November 1^st?
A radio-active substance’s mass has a continuous decay rate of per day. Assume an initial mass of grams.
1. What would be the mass of the substance after $d$ days?
2. What would be the mass of the substance after $y$ years?

Problem set

Dr.Punshen-Smith is experimenting with bacteria in his lab. He starts with a population of $10000$ bacteria. He observes that the bacteria population follows the model $10000\times e^{0.5t}$ , where $t$ is time in days. How long does it take for the bacteria to double in Dr.Punshen-Smith’s experiment?
The bacterial population in Prof.Riehl’s lab experiment has a continuous growth rate of $25\%$ per day. How long does it take for the population to double?
The viral population in Prof.Benedetto’s lab experiment has a continuous growth rate of $100\%$ per month. How long does it take for the population to double?
A particular radio-active substance has a continuous decay rate of $25\%$ per day. What is the half-life of the radio-active substance?
A radio-active rock on a distant planet was found to have a continuous decay rate of $50\%$ per year. What would be the half-life of the radio-active substance?
Dr.Bedrossian is experimenting with bacteria in his lab. He starts with a population of $10000$ bacteria. He observes the bacteria doubles every $5$ hours. Give a continuous growth rate model for the bacteria population in Dr.Bedrossian’s experiment.
Dr.Blumenthal is experimenting with bacteria in his lab. He starts with a population of $1000$ bacteria. He observes the bacteria triples every $6$ hours. Give a continuous growth rate model for the bacteria population in Dr.Blumenthal’s experiment.
A radio-active substance’s half-life is $4$ years. Assuming an initial mass of $M$ grams, give a continuous growth rate model for the mass.

Problem set

Michelle opens an account at Liberty bank where the account balance is modelled as $P\times e^{0.3t}$ . What is the compounding period and what is the annual interest rate?
Joe opens an account at Union bank where the account balance is modelled as $P\times e^{0.25t}$ . What is the compounding period and what is the annual interest rate?
Anthony opens an account at Millenium bank with a deposit of $\$1000$ . The account is such that it grows its balance continuously at a rate of $12\%$ per year. Assuming Anthony does not do any withdrawals or deposits for this account, give a function to model the account balance $t$ years after the account was opened.
Sonia opens an account at a bank that compounds continuously at a rate of $100\%$ per year. How long does it take for Sonia’s account balance to double?
A bank employs a continuous compounding scheme on one of its financial instruments and guarantees that the instrument gives doubling returns every $5$ years. What is the annual interest rate? Assuming an initial balance of $B$ , give a continuous compounding model for the returns as a function of the number of years elapsed.

Miscellaneous

Exercises

Problem set

Answer if each of the following functions is increasing or decreasing.

$10\times 1.1^x$
$10\times 2^x$
$10\times 0.9^x$
$10\times 0.5^x$
$0.1\times 2^x$
$-10\times 1.1^x$
$-10\times 0.5^x$

Problem set

Which of the following relations are functions? Assume the relations are from the set $\mathbb{R}$ to the set $\mathbb{R}$ . In case they are not functions, would restricting the sets in some way make them functions? Identify domain, codomain and range for the each of the functions.

$2^x$
$1^x$
$\left(0.2\right)^x$
$\left(\sqrt{2}\right)^x$
$\left(-2\right)^x$
$\left(2^x\right)^2$