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 exactly, 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, the 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 initially there are $150$ 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 there be in the culture after $4$ hours?

Calculate analytically the coordinates of the points of intersection with the x-axis and the y-axis 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 of base $2$ and solve:

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

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

$\$5000$ is invested in a bank deposit that offers an annual compound interest 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 a suitable change of variable:

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

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 (time it takes for its initial mass to halve) of $8$ days. If we start with 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 by applying logarithms:

$4^{x-1} = 15$

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

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, the range and the equation of the horizontal asymptote of $g(x)$.

The market value of an industrial machine decreases exponentially with time. It was bought for $\$40000$ and it is known that each year it loses $15\%$ of its value with respect 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?

• How much exact time must elapse for its value to halve?

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 time elapsed in days since 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 needed for half 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.

• The points of intersection with the coordinate axes.

• 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 throughout $\mathbb{R}$:

Solve the following exponential equation, taking out a common factor after applying the properties of powers:

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

According to Newton's law of cooling, the temperature of an object at a 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 start 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 $g(x)$. Next, calculate the exact area of the triangle whose vertices are this point of intersection and the two points where the curves intersect the line $y = e$.