Exponential Functions

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

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 $a$.

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 $y = \left(\frac{1}{3}\right)^x$.

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

• a) $2^x = 64$

• b) $3^x = \frac{1}{9}$

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.

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 $b$.

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

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.

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

• a) $5^{2x-1} = 125$

• b) $2^{x+3} = \frac{1}{4}$

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

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

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

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

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.

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 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.

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.