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 $40\,000$ pesos 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 $f(x) = 0.5^x$.

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 $100\,000$ pesos in $5$ years. If their bank offers a deposit with a compound interest rate of $3 \%$ per year, 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 cell phone newly released on the market is $16\,000$ pesos. Due to technological obsolescence, its value depreciates by $20 \%$ each year. Write the function modeling the price of the phone in terms of the years elapsed and calculate what its value will be after $4$ years.

The growth of a certain insect pest follows an exponential model of the form $N(t) = k \times a^t$, 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 $200\,000$ pesos 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 $T(t) = 20 + 70 \times 0.8^t$, 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 $60\,000$ pesos 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:

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