Power Functions

Identify which of the following algebraic expressions correspond to a power function:

• $f(x) = 3x^2$

• $g(x) = 2^x$

• $h(x) = \frac{5}{x^3}$

• $i(x) = x^{1/2}$

Given the power function $f(x) = 2x^4$, calculate the value of its images for $x = -2$, $x = 0$ and $x = 3$.

Find the value of the coefficient $k$ of the power function $f(x) = k \cdot x^3$ knowing that its graph passes through the point with coordinates $(2, 40)$.

Knowing that the graph of a power function of the form $f(x) = x^n$ passes through the point $(3, 81)$, find the value of the exponent $n$.

Indicate the domain of definition and determine the type of symmetry (even, odd or neither) of the power function $f(x) = x^6$.

Indicate the domain of definition and the equations of the vertical and horizontal asymptotes of the power function $f(x) = x^{-2}$.

Given the function $f(x) = -3x^{-3}$, determine its domain of definition, its range and study analytically if it presents any type of symmetry (even or odd).

Write the algebraic expression of the function that associates each positive real number with its cube root, expressing it as a power with a fractional exponent.

Given the power function with a fractional exponent $f(x) = x^{2/3}$:

• Determine its domain of definition.

• Calculate the exact values of $f(8)$ and $f(-8)$.

• Reason analytically whether the function is even, odd or neither.

Observe the following graph showing two power functions in the first quadrant.

Knowing that the analytical expressions correspond to $y = x^2$ and $y = x^3$, identify which is $f(x)$ and which is $g(x)$. From the graph and the expressions, determine in which strictly positive interval the cubic function takes smaller values than the quadratic function.

The following graph shows three power functions whose expressions are $y = x^{-1}$, $y = x^{-2}$ and $y = x^{1/2}$.

Match each curve (A, B and C) with its corresponding analytical expression, justifying your answer based on the domain, parity and asymptotic behavior of each one.

Determine analytically the points of intersection between the cubic function $f(x) = 8x^3$ and the inverse proportion function $g(x) = x^{-1}$.

A power function is given by the expression $f(x) = a \cdot x^n$. Find the exact values of the coefficient $a$ and the exponent $n$ knowing that the graph passes through the points $(1, 5)$ and $(2, 40)$.

Find the analytical expression of the power function $f(x) = a \cdot x^n$ knowing that its graph passes through the points $(2, 2)$ and $(8, 16)$. Express the exponent $n$ as an irreducible fraction.

The period $T$ of a simple pendulum (the time it takes to complete one full oscillation, in seconds) can be approximated by the power function $T(l) = 2 \cdot l^{1/2}$, where $l$ is the length of the pendulum in feet. Calculate the period of a pendulum $0.25$ ft long and find out what length a pendulum must have for its period to be exactly $4$ seconds.

The volume of a sphere as a function of its radius is calculated using the formula $V(r) = \frac{4}{3}\pi r^3$. Rearrange for the radius $r$ to express it as a power function dependent on the volume $V$. Use this new function to calculate the approximate radius of a sphere that has a volume of $1000$ in$^3$ (take $\pi \approx 3.14$).

The production cost of an electronic component decreases as the number of units manufactured increases according to the power function $C(x) = 5000 \cdot x^{-1/2}$, where $x$ is the quantity of components and $C(x)$ is the unit cost in dollars. Calculate the unit cost of producing $100$ components and determine how many components must be manufactured in total for the unit cost to be reduced to $\$25$.

Express the area $A$ of a circle as a function of its radius $r$ using a power function. What is the degree of this power function?

The light intensity $I$ received by an object is inversely proportional to the square of the distance $d$ to the light source, which is modeled by the power function $I(d) = k \cdot d^{-2}$. If at a distance of $2$ feet the intensity is $100$ lux, find the complete expression of the function and calculate the intensity the object will receive at a distance of $5$ feet.

The kinetic energy of a moving body is given by the formula $E_c(v) = \frac{1}{2} m v^2$. If the mass of a vehicle is $1200$ pounds, express its kinetic energy as a power function dependent on its speed $v$ (in ft/s) and calculate the energy it has when traveling at $20$ ft/s.

The basal metabolic rate $T$ (in kcal/day) of mammals is modeled as a function of their body mass $M$ (in pounds) using Kleiber's power function: $T(M) = 70 \cdot M^{3/4}$. Calculate the basal metabolic rate of a dog weighing $16$ pounds. Next, find the approximate mass of an animal whose metabolic rate is $7000$ kcal/day.

Kepler's third law states that the square of the orbital period $T$ of a planet is directly proportional to the cube of its mean distance to the Sun $R$, that is, $T^2 = k \cdot R^3$. If for the Earth it is true that $T = 1$ year and $R = 1$ AU (Astronomical Unit), express the period $T$ as a power function dependent on $R$. Use this function to calculate the orbital period of Jupiter, knowing that its mean distance to the Sun is $5.2$ AU.

The aerodynamic drag force experienced by a cyclist is proportional to the square of their speed, which is expressed as $F(v) = k \cdot v^2$. At a speed of $10$ ft/s, the measured force is $30$ N. Prove analytically by what exact factor the drag force is multiplied if the cyclist doubles their speed (going from a generic speed $v_0$ to $2v_0$) and calculate the new force when traveling at $20$ ft/s.