Exercises: Exponential function

When studying translations of basic functions, a student claims that the graph of $y = 3^{x+2}$ is obtained by shifting the original graph of $y = 3^x$ two units to the right. Correct this statement by indicating the actual direction of the shift.

When simplifying the expression of an exponential function $f(x) = 2^x \cdot 2^3$, a student obtains $f(x) = 4^{x+3}$ as the result. Explain which power rule they applied incorrectly when manipulating the bases, and write the correct simplified expression.

State which of the following algebraic expressions correspond to an exponential function and justify your answer:

1. a) $f(x) = x^3$
2. b) $g(x) = 3^x$
3. c) $h(x) = 3x$
4. d) $i(x) = \left(\frac{1}{4}\right)^x$
5. e) $j(x) = 1^x$

Given the exponential function $f(x) = 4^x$, calculate the values of the function for $x = -2$, $x = -1$, $x = 0$, $x = 1$ and $x = 2$. Express the results as irreducible fractions when necessary.

Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base:

1. a) $f(x) = 1,8^x$
2. b) $g(x) = \left(\frac{2}{5}\right)^x$
3. c) $h(x) = 0,3^x$
4. d) $i(x) = \left(\frac{7}{4}\right)^x$

The graph of an exponential function of the form $f(x) = a^x$ passes through the point $(3, 125)$. Determine the value of the base $a$ and write the algebraic expression of the function.

In a laboratory, the growth of a bacterial culture is being studied. Initially, $50$ bacteria are placed in a Petri dish, and it is observed that their number doubles every hour. Write the function $P(t)$ that gives the number of bacteria as a function of time $t$ (in hours) and calculate how many there will be after $4$ hours.

Look at the graphs of the exponential functions shown in panels A and B. One corresponds to $f(x) = 3^x$ and the other to $g(x) = \left(\frac{1}{3}\right)^x$. Assign each expression to its corresponding graph and explain the geometric relationship between them.

Determine the domain, range and equation of the horizontal asymptote of the exponential function $f(x) = 2,5^x$. Does this function cross the x-axis at any point? Give reasons for your answer.

Complete a table of values for the function $f(x) = 5^{-x}$ using integer values of $x$ from $-2$ to $2$. Write an exponential function equivalent to $f(x)$ but with a positive exponent.

Type: SOMEONE ELSE'S ERROR. Marcos wants to calculate how a delivery van in Madrid whose initial price was €$24\,000$ loses value. At the dealership, they tell him that the book value of the vehicle depreciates by $15\%$ each year, following the exponential function $V(t) = 24\,000 \cdot 0{,}85^t$, where $t$ is the elapsed time in years. Marcos carries out the following reasoning in his notebook: «If it loses $15\%$ each year, after $7$ years it will have lost a total of $7 \cdot 15\% = 105\%$ of its original value. Therefore, at $7$ years the vehicle will be worth negative $5\%$, that is, $V(7) = -1200$ €».\na) Identify and explain the conceptual error that Marcos makes when accumulating the percentage losses.\nb) Explain, using the definition of an exponential function and its horizontal asymptote, why the value obtained through $V(t)$ can never become negative or equal zero.\nc) Correct Marcos's calculation by determining the estimated real value of the van after $7$ years (round the result to the nearest euro cent).

Type: MODEL. According to data from the Instituto Nacional de Estadística (INE), a rural municipality in Soria had $1250$ inhabitants as of January 1, 2018. Due to depopulation and demographic aging, its census steadily decreases at an approximate rate of $4\%$ per year.\na) Write the analytic expression of the exponential function $P(t)$ that models the estimated number of inhabitants as a function of the elapsed years $t$ since 2018.\nb) Identify the base $a$ of your function. Justify why it is a decreasing function by relating it to the value of this base.\nc) Calculate the projected population for the year 2026, rounding to the nearest whole number of people. Will the mathematical function ever predict exactly $0$ inhabitants? Justify your answer by comparing what the horizontal asymptote of the mathematical model indicates with what will inevitably happen in the human reality of the town.

A family deposits $6000$ euros into a savings account offering an annual compound interest rate of $2,5\%$. Write the function $C(t)$ expressing the accumulated capital after $t$ years. What will the account balance be after $8$ years if no transactions are made? Round the result to two decimal places.

Starting from the function $f(x) = 2^x$, explain what geometric transformations (translations) must be applied to obtain the graph of the function $g(x) = 2^{x-3} + 4$. State the equation of the horizontal asymptote of the function $g(x)$.

The price of a technological component depreciates over time according to an exponential function of the form $V(t) = k \cdot a^t$, where $t$ is measured in years. If the component cost $800$ euros when new ($t=0$) and after $2$ years its value is $450$ euros, calculate the exact values of $k$ and $a$. What will its value be after $4$ years?

A radioactive substance loses half of its mass every $15$ days. If initially there is a sample of $200$ grams, write the function $M(t)$ that models the remaining mass in terms of time $t$ (in days). How many grams will remain exactly after $60$ days?

Calculate the average rate of change of the exponential function $f(x) = 3^x$ in the intervals $[1, 2]$ and $[2, 3]$. What do you observe when comparing both results? What does this indicate about the rate of growth of the exponential function compared to a linear function?

The following graph corresponds to an exponential function of the form $f(x) = 2^{x-p} + q$. By observing the horizontal asymptote and the marked points on the curve, determine the values of the parameters $p$ and $q$, and write the complete algebraic expression of the function.

Fake news spreads on a social network following an exponential model $N(t) = 15 \cdot 1,4^t$, where $N(t)$ is the number of people who know the news and $t$ is the time elapsed in hours. How many people knew the news at the start of the measurement? How many will know it after $6$ hours? (Round to the nearest whole number).

Find analytically the coordinates of the axis intercepts of the exponential function $f(x) = 3^{x+2} - 27$.

Type: INVERSE PROBLEM. In a clinical trial on the elimination of an analgesic from the body, it is known that the amount of residual active ingredient in the blood $C(t)$ (in milligrams) follows an exponential function of the form $C(t) = C_0 \cdot a^t$, where $t$ represents the time in hours since ingestion. Due to a computer registration error, the data for parameters $C_0$ and $a$ have been lost, but two exact measurements are preserved in the medical record: at the initial instant ($t = 0$) the amount was $600$ mg, and at exactly $3$ hours the remaining amount was $75$ mg.\na) Reasonably determine the value of the initial parameter $C_0$ and find the exact value of the base $a$. Then, write the complete formula for the function $C(t)$.\nb) Using the value of $a$, state what percentage of the drug present in the body is eliminated each hour.\nc) If the medication ceases to have a therapeutic effect when the residual amount drops below $10$ mg, carry out a justified trial-and-error by evaluating the function at integer values of $t$ to find between which two consecutive hours the treatment ceases to be effective.

Type: MODEL. The State Meteorological Agency (AEMET) records a standard barometric pressure of $1013$ hPa (hectopascals) at sea level in the Canary Islands. As altitude increases, air density decreases and atmospheric pressure drops by approximately $12\%$ for each kilometer of elevation, following the function $P(h) = 1013 \cdot 0{,}88^h$, where $h$ is the altitude in kilometers.\na) The Teide National Park visitor center is located at an altitude of $2$ km, and the summit of the volcano is at about $3{,}7$ km. Calculate the approximate atmospheric pressure at both locations, rounding to the nearest whole hPa.\nb) When climbing from sea level ($h = 0$) to an altitude of $1$ km, the pressure drops by more than $120$ hPa, whereas when going from $3$ km to $4$ km of altitude, the pressure drops by less than $85$ hPa. Justify this phenomenon using the graphical properties of a decreasing exponential function.\nc) An oceanographic team dives $1$ km below sea level off the coast of El Hierro. Does it make mathematical sense to evaluate $P(-1)$ in the given function? Does it make physical sense to interpret that result as the actual pressure underwater? Justify your answer by distinguishing between the abstract mathematical domain and the range of validity of a real-world model.

Determine and justify for which values of the parameter $m$ the function $f(x) = \left(\frac{2m - 5}{3}\right)^x$ represents a strictly decreasing exponential function. Take into account the algebraic conditions that the base of such an exponential function must satisfy.

In an ecosystem, two species of insects coexist. The population of species A is given by $P_A(t) = 500 \cdot 4^t$, and that of species B by $P_B(t) = 32000 \cdot 0,5^t$, where $t$ is time in months. At what time $t$ will both populations have exactly the same number of individuals? Solve this by setting up and solving the corresponding exponential equation.

Look at the graph of the exponential function, whose algebraic expression has the form $f(x) = k \cdot a^x + c$. Identify the equation of the horizontal asymptote to deduce the value of $c$, and use the coordinates of the marked points in the figure to set up a system of equations to calculate the parameters $k$ and $a$. Write the final expression of the function.

Study the continuity of the following piecewise function at the point $x = 2$:

$f(x) = \begin{cases} 3^{x-1} + 2 & \text{if } x \le 2 \\ m \cdot x - 1 & \text{if } x > 2 \end{cases}$

Find the exact value of the parameter $m$ so that the function is continuous throughout its domain $\mathbb{R}$.

According to Newton's law of cooling, the temperature of an object placed in a room at constant temperature is modelled by an exponential function. A freshly made cup of coffee follows the function $T(t) = 22 + 68 \cdot 0,85^t$, where $T$ is the temperature in degrees Celsius and $t$ is the time in minutes.

1. a) What is the initial temperature of the coffee?
2. b) What is the room's ambient temperature? (Hint: analyse the trend of the function as the time $t$ becomes extremely large).
3. c) What temperature will the coffee be after $10$ minutes?

Given the functions $f(x) = 2^x$ and $g(x) = 4 - x^2$, reason graphically how many intersection points they have. Then, check by trial and error between which consecutive integers the abscissas of these intersection points lie, testing with small integer values.

A marketing company offers two pay plans to its sales representatives. Plan A consists of a basic salary of $1000$ euros in the first month, plus a fixed increase of $200$ euros for each additional month of seniority. Plan B offers an initial salary of $100$ euros in the first month, but the amount is multiplied by $1,5$ each month. Write the functions $S_A(t)$ and $S_B(t)$ (where $t$ is the month number) and determine, by calculating the values month by month, from which month onwards Plan B becomes strictly more profitable than Plan A.

The mass $M$ in grams of a radioactive isotope decreases over time $t$ (in years) according to the expression $M(t) = M_0 \cdot 3^{-t/20}$, where $M_0$ is the initial mass. If it is known that after $40$ years there are exactly $15$ grams of isotope remaining, calculate the initial mass $M_0$. How many grams will remain when $60$ years in total have elapsed?

Type: CHOOSE THE METHOD. A bicycle and van delivery company in Valencia purchases a commercial vehicle for €$32\,000$. To calculate its balance sheet over the coming years, the manager is deciding between two accounting depreciation methods:\n- Linear method: The vehicle loses a fixed amount of €$3200$ each year: $f(t) = 32\,000 - 3200 \cdot t$.\n- Exponential method: The vehicle loses $15\%$ of its book value from the previous year: $g(t) = 32\,000 \cdot 0{,}85^t$.\na) Calculate the value that each method assigns to the vehicle after $3$ years of use and after $10$ years of use.\nb) Examine what would happen with the linear method after $11$ years of use. Why does the linear model lose mathematical and economic validity in the long term? Explain which property of the exponential function (range and asymptotes) makes the exponential model conceptually more coherent for describing prolonged losses of value.\nc) Over the vehicle's first $4$ years of life, which of the two models reflects a faster depreciation? If company management decides to apply a criterion of maximum financial prudence (reflecting the lowest possible value in the early years so as not to overvalue their assets), mathematically justify which of the two methods they should choose.

Type: ESTIMATE AND CHECK. In the Nuclear Medicine department of a public hospital, the radioisotope Technetium-99m is used for cardiac scans. Its mass decays exponentially with a half-life of $T = 6$ hours, following the formula $M(t) = M_0 \cdot \left(\frac{1}{2}\right)^{t/6}$, where $t$ is the time in hours and $M_0$ is the initial activity in MBq (megabecquerels). At 08:00 in the morning, a dose with $M_0 = 800$ MBq is received in the laboratory.\na) Estimate: Without performing calculations with decimals or using a calculator, mentally calculate how many complete half-life periods fit into $24$ hours (until 08:00 the next day) and estimate whether the remaining activity will be above $100$ MBq, between $40$ and $60$ MBq, or below $20$ MBq. Justify your deduction in writing.\nb) Check: Apply the formula by calculating the exact value of $M(24)$ using powers with a fractional base and check whether your reasoned estimate from the previous part was correct.\nc) The hospital safety protocol stipulates that a used vial cannot be disposed of with ordinary healthcare waste until its residual activity has dropped below $1\%$ of the initial activity ($8$ MBq). First, estimate how many half-lives (successive powers of $1/2$) are needed as a minimum to reduce any quantity below $1\%$, and then calculate the exact minimum number of hours that the waste must remain in the lead-shielded isolation chamber.