Exercises: Exponential functions

Given the exponential function $f(x) = 2^x$, calculate the outputs for the values $x = -2$, $x = 0$ and $x = 3$.

State, giving reasons for your answer, which of the following exponential functions are increasing and which are decreasing:

• a) $f(x) = 1,2^x$

• b) $g(x) = 0,7^x$

• c) $h(x) = \left(\frac{4}{3}\right)^x$

• d) $i(x) = \left(\frac{2}{5}\right)^x$

Look at graphs A and B shown in the following figure. Match each of them with its corresponding algebraic expression: $y = 3^x$ or $y = \left(\frac{1}{3}\right)^x$.

A family deposits $2000$ € into a bank account offering a compound interest rate of $4 \%$ per annum. What capital will they have after $2$ years if they do not withdraw any money from the account?

The number of followers of an account that has just gone viral triples every day. If it initially has $100$ followers, write the function relating the number of followers to the time elapsed in days, and calculate how many followers there will be after $3$ days.

Find the value of $x$ in the following basic exponential equations:

• a) $2^x = 64$

• b) $3^x = \frac{1}{9}$

The graph of an exponential function of the form $f(x) = a^x$ passes through the point $(3, 8)$. Calculate the value of the base $a$.

Determine the domain, the range and the y-intercept of the exponential function $f(x) = 0,5^x$.

Marcos is analyzing the function $f(x) = 2^{x+1} - 8$. On his exam, he wrote the following conclusion: “Since it is an exponential function, its horizontal asymptote is always the line $y = 0$, its range is $(0, +\infty)$ and, therefore, the graph never intersects the $x$-axis nor does it take negative values”.

1. a) Identify the two conceptual errors Marcos made regarding the translations of an exponential function.
2. b) Reasonably explain why these errors occur and correct them, stating the correct equation of the horizontal asymptote and the actual range of $f(x)$.
3. c) Prove that the function does intersect the $x$-axis by calculating the exact coordinates of that point of intersection.

Determine the analytical expression of an exponential function of the form $f(x) = a^x + q$ from the following information:

1. Its graph approaches the horizontal line $y = -3$ indefinitely when the values of $x$ are very negative, without ever touching it.

2. The graph passes through the point $P(2, 13)$.

Once the parameters $a$ and $q$ have been calculated, explain whether the function is increasing or decreasing by observing the value of the obtained base, and find the coordinates of its $y$-intercept.

Plot the graph of the function $f(x) = 3^{x-2} + 1$ by first making a table of values. State the equation of its horizontal asymptote and calculate the coordinates of its y-intercept.

You can do it on paper and send a photo with your answer.

Solve the following exponential equations by making the bases equal on both sides of the equation:

• a) $5^{2x-1} = 125$

• b) $2^{x+3} = \frac{1}{4}$

An investor wishes to obtain a final capital of $5000$ € in $5$ years' time. If their bank offers a deposit with a compound interest rate of $3 \%$ per annum, what initial capital must they invest today to achieve their goal? (Round the answer to the nearest cent).

A sample of a radioactive isotope has an initial mass of $200$ grams. It is known that its mass halves every $10$ years. Write the algebraic expression for the function giving the remaining mass in terms of time (in years) and calculate how many grams will remain after exactly $30$ years.

The following graph corresponds to an exponential function of the form $f(x) = a^x + b$. Look at its key features, such as the horizontal asymptote and the points through which it passes, and determine the values of the parameters $a$ and $b$.

Solve the following exponential equation by first taking out a common factor:

$3^x + 3^{x+1} = 36$

Find algebraically the exact coordinates of the x-intercepts and y-intercepts of the exponential function $f(x) = 2^{x+1} - 8$.

The price of a high-end mobile phone newly released on the market is $800$ €. Due to technological obsolescence, its value depreciates by $20 \%$ each year. Write the function modelling the price of the phone in terms of the years elapsed and calculate what its value will be after $4$ years.

In a hospital, a patient is given an $80\text{ mg}$ dose of a drug. The mass $M(t)$ of the drug remaining in their body (in milligrams) after $t$ hours is described by the exponential function $M(t) = 80 \cdot \left(\frac{1}{2}\right)^{t/4}$.

1. a) Without calculating the exact value using the formula, reasonably estimate between which two consecutive tens of milligrams the amount of drug will be after 10 hours have elapsed (hint: mentally calculate the exact amount remaining at 8 and 12 hours).
2. b) Now calculate the exact numerical value of $M(10)$, rounding to the nearest hundredth, and check whether your estimate from the previous part was accurate.
3. c) Based on the properties of the exponential function (its horizontal asymptote and its range), explain whether there will be any moment when the patient's body mathematically has exactly $0\text{ mg}$ of the drug according to this model.

According to demographic data inspired by the censuses of the National Statistics Institute (INE), a rural municipality had $1200$ inhabitants at the beginning of 2015. Due to aging and emigration, its population decreases at a constant rate of $5\,\%$ each year.

1. a) Write the formula for the function $P(t)$ that expresses the number of inhabitants as a function of the time $t$ elapsed in years since 2015. Justify whether the function is increasing or decreasing by analyzing the value of its base.
2. b) Calculate the projected population for this municipality at the beginning of the year 2023, rounding to the nearest whole number of people.
3. c) A municipal spokesperson states in the press: “By losing $5\,\%$ each year, in 20 years we will have lost $100\,\%$ of the population and the town will have $0$ inhabitants”. Justify mathematically why this linear reasoning is flawed and calculate what percentage of the initial 2015 population will actually remain after those 20 years.

The growth of a certain insect pest follows an exponential model of the form $N(t) = k \cdot a^t$, where $N(t)$ is the number of insects and $t$ is the time in weeks. Knowing that initially ($t=0$) there were $5$ insects and that after $2$ weeks there were $45$ insects, calculate the values of the constants $k$ and $a$. Then, determine how many insects there will be after $4$ weeks.

Two siblings receive an inheritance of $10000$ € each. The first decides to invest their money at a simple interest rate of $5 \%$ per annum, while the second invests it at a compound interest rate of $4 \%$ per annum. Write the functions describing the capital of each sibling over time. Who will have more money after $10$ years? And after $20$ years?

The population of city A is currently $100000$ inhabitants and grows at a constant rate of $4 \%$ per annum. The population of city B is $150000$ inhabitants and decreases at a rate of $2 \%$ per annum. Write the population functions for both cities. Check, using trial and error on a calculator (or by applying logarithms), in which exact year the population of city A will exceed that of city B.

Given the modulus function $f(x) = |3^x - 9|$:

• a) Write it as a piecewise function, first calculating the point at which the expression inside changes sign.

• b) Determine its axis intercepts.

• c) State the intervals in which the function is increasing and those in which it is decreasing.

According to Newton's law of cooling, the temperature of a cup of tea in a room is modelled by the function $T(t) = 20 + 70 \cdot 0,8^t$, where $T(t)$ is the temperature in degrees Celsius and $t$ is the time in minutes elapsed since it was poured.

• a) What was the initial temperature of the tea?

• b) What is the ambient temperature of the room? (Reason what happens as the time $t$ becomes very large).

• c) Set up the equation to calculate at which minute the temperature of the tea will be exactly $55$ ºC and solve it.

Pharmacokinetics studies how a medication is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the bloodstream decreases exponentially such that every $6$ hours the amount halves.

• a) Write the function $C(t)$ expressing the milligrams of drug in the bloodstream in terms of time $t$ measured in hours.

• b) What amount of drug will remain in the patient's bloodstream after a full day ($24$ hours)?

A bank offers an investment product with a nominal annual interest rate of $6 \%$.

• a) Write the function modelling the final capital if $3000$ € is invested and interest is paid annually (annual compound interest).

• b) Write the function if interest is paid monthly (monthly compound interest).

• c) Calculate the exact difference in capital obtained between the two options after $5$ years.

Solve the following exponential equation by applying the laws of indices and taking out a common factor:

$2^{x-1} + 2^{x-2} + 2^{x-3} = 56$

To solve the exponential equation $4^x - 3 \cdot 2^{x+1} - 16 = 0$, three different solving strategies are proposed:

• Strategy A: Attempt to factor out $2^x$ directly from the three terms just as they are written.

• Strategy B: Attempt to equate the bases directly without transforming the structure of the equation.

• Strategy C: Express all powers in base 2, simplify the coefficients, and apply the substitution $t = 2^x$ to transform the expression into a quadratic equation.

1. a) Explain in detail why Strategies A and B are not viable for this equation.
2. b) Apply Strategy C to solve the equation step by step. During the process you will obtain two possible values for $t$; reasonably justify which one you must discard based on the range of the exponential function $y = 2^x$.
3. c) Find the final value of $x$ and verify that it satisfies the original equation by substituting it into it.

In a microbiology laboratory, the simultaneous growth of two different bacterial colonies, Strain A and Strain B, is analyzed under controlled culture conditions:

• Strain A initially has 150 bacteria and its population doubles every 3 hours, following the model $N_A(t) = 150 \cdot 2^{t/3}$ ($t$ measured in hours).

• Strain B starts with a larger population, with 1200 bacteria, but reproduces more slowly, doubling every 6 hours: $N_B(t) = 1200 \cdot 2^{t/6}$.

1. a) Write each function in the form $N(t) = N_0 \cdot b^t$ by determining the hourly base $b$ of each one. Based on the value of these bases, justify which of the two strains grows at a higher percentage rate per hour.
2. b) Even though Strain B starts with 8 times more bacteria than Strain A, will Strain A eventually catch up to it? Set up the equation $N_A(t) = N_B(t)$ and solve it using exclusively the properties of powers to find at what exact hour they will have an identical number of bacteria.
3. c) Calculate the number of bacteria in each strain after 24 hours have elapsed and find the difference between the two. Interpret whether the result obtained agrees with what was reasoned in the previous sections.