Exercises: Exponential functions

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

Indicate, reasoning 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) $i(x) = \left(\frac{2}{5}\right)^x$

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

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

A culture of bacteria is studied in a laboratory. Initially there are $100$ bacteria and it is observed that their number triples every hour. Write the function that relates the number of bacteria to the time elapsed in hours, and calculate how many bacteria there will be after $3$ hours.

Find the value of $x$ in the following elementary exponential equalities:

• a) $2^x = 64$

• b) $3^x = \frac{1}{9}$

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

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

Plot the function $f(x) = 3^{x-2} + 1$ graphically by first drawing up a table of values. Indicate the equation of its horizontal asymptote and calculate the coordinates of its point of intersection with the y-axis.

Solve the following exponential equations by equating the bases on both sides of the equality:

• 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 them a deposit with a compound interest of $3 \%$ per year, what initial capital must they invest today to achieve their goal? (Round the result to the nearest cent).

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

The following graph corresponds to an exponential function of the form $f(x) = a^x + b$. Observe its characteristic elements, such as the horizontal asymptote and the points it passes through, 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, analytically, the exact coordinates of the points of intersection with the x-axis and the y-axis of the exponential function $f(x) = 2^{x+1} - 8$.

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

The evolution of a certain insect plague 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. Given that at the initial instant ($t=0$) there were $5$ insects and that at $2$ weeks there were $45$ insects, calculate the values of the constants $k$ and $a$. Then, determine how many insects there will be at $4$ weeks.

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

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

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

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

• b) Determine its points of intersection with the coordinate axes.

• c) Indicate in which intervals the function is increasing and 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 Fahrenheit and $t$ is the time in minutes elapsed since it was served.

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

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

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

Pharmacokinetics studies how a medicine is eliminated from the human body. A patient is administered an initial dose of $50$ mg of a drug. The amount in the blood decreases exponentially so that every $6$ hours the amount is halved.

• a) Write the function $C(t)$ that expresses the milligrams of the drug in the blood in terms of the time $t$ measured in hours.

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

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

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

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

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

Solve the following exponential equation by applying the properties of exponents and taking out a common factor:

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