Exercises: Exponential functions

Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using roots or fractions if necessary.

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

• $f(x) = 4^x$

• $g(x) = \left(\frac{1}{3}\right)^x$

• $h(x) = 0.8^x$

• $i(x) = e^x$

Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$.

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

$5^{2x-1} = 125$

A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a function of time $t$ (expressed in hours). How many bacteria will be in the culture after $4$ hours?

Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$.

Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0.5^x$, $h(x) = 3^x$ and $i(x) = -2^x$.

Express both sides of the equation as powers with base $2$ and solve:

$8^x = \frac{1}{4}$

Find the algebraic expression for an exponential function of the form $f(x) = k \times a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.

An amount of $300\,000$ pesos is invested in a bank deposit offering an annual compound interest rate of $3\%$.

• Write the function that determines the accumulated capital $C(t)$ after $t$ years.

• Calculate the total capital that will have accumulated after $10$ years.

Solve the following exponential equation by making an appropriate substitution:

$9^x - 10 \times 3^x + 9 = 0$

Determine the domain of definition of the following function:

$f(x) = \frac{x+2}{e^x - e^2}$

A radioactive isotope has a half-life (the time it takes for its initial mass to be reduced by half) of $8$ days. Starting from an initial sample of $200$ grams:

• Write the function $M(t)$ that gives the remaining mass of the isotope as a function of time $t$ in days.

• What mass of this isotope will remain after $24$ days?

Solve the following exponential equation using logarithms:

$4^{x-1} = 15$

Express the result in exact form using logarithms and also provide an approximation rounded to two decimal places.

Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function $g(x) = 2^{x-3} + 1$. Determine the domain, range and the equation of the horizontal asymptote of $g(x)$.

The market value of an industrial machine decreases exponentially over time. It was bought for $2\,400\,000$ pesos and it is known that each year it loses $15\%$ of its value relative to the immediately preceding year.

• Write the function $V(t)$ that gives the value of the machine as a function of the years elapsed.

• What will its value be after $5$ years?

• Exactly how much time must elapse for its value to be halved?

The spread of a virus in an isolated population of $10000$ inhabitants follows a logistic growth model given by the function:

$P(t) = \frac{10000}{1 + 99 e^{-0.5t}}$

where $P(t)$ is the number of infected people and $t$ is the elapsed time in days from the start of the outbreak.

• Calculate the initial number of infected people at time $t = 0$.

• Determine how many people will be infected after $10$ days.

• Calculate the exact time required for half of the population to be infected.

Solve the following system of exponential equations:

Given the function $f(x) = e^{\frac{2x}{x-1}}$, carry out an analytical study to determine:

• Its domain of definition.

• Its axis intercepts.

• The equations of its horizontal and vertical asymptotes, if it has any.

Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous on all of $\mathbb{R}$:

Solve the following exponential equation by factorising out a common factor after applying index laws:

$2^{x+2} + 2^{x-1} - 2^{x-2} = 34$

According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is the ambient temperature and $T_0$ is the initial temperature of the object. A cup of coffee at $90^\circ\text{C}$ is left in a room kept at a constant temperature of $20^\circ\text{C}$. If after $10$ minutes the temperature of the coffee has dropped to $60^\circ\text{C}$:

• Calculate the exact value of the cooling constant $k$.

• Determine the total time elapsed from the beginning for the coffee to reach a temperature of $30^\circ\text{C}$.

Determine analytically the intervals of constant sign (that is, the intervals where the function is strictly positive and where it is strictly negative) of the function:

$f(x) = (x^2 - 5x + 6)e^{-x}$

The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$.

Determine analytically the coordinates of the point of intersection between the functions $f(x)$ and $g(x)$. Then, calculate the exact area of the triangle whose vertices are this point of intersection and the two points where the curves intersect the line $y = e$.