Power Functions

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

• $f(x) = 3x^2$

• $g(x) = 2^x$

• $h(x) = \frac{5}{x^3}$

• $i(x) = x^{1/2}$

Write the algebraic expression for the function that assigns each positive real number its cube root, expressing it in index form with a fractional exponent.

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

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

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

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

Look at the following graph showing two power functions in the first quadrant.

Knowing that the algebraic 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 values less than the quadratic function.

Determine algebraically the points of intersection between the graphs of the power functions $f(x) = x^2$ and $g(x) = x^4$.

State 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 algebraically whether it exhibits any type of symmetry (even or odd).

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 $f(-8)$.

• State with algebraic reasoning whether the function is even, odd or neither.

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 algebraic expression, justifying your answer based on the domain, parity and asymptotic behavior of each one.

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 $n$.

A power function is given by the expression $f(x) = a \times x^n$. 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 algebraic expression for the power function $f(x) = a \times x^n$ knowing that its graph passes through the points $(2, 2)$ and $(8, 16)$. Express the exponent $n$ as an irreducible fraction.

The radius $r$ (in cm) of a circular oil slick spreading on water grows with time $t$ (in minutes) according to the power function $r(t) = 2 \times t^{1/2}$. Calculate the radius of the slick after $0.25$ minutes and find out how much time must pass for the radius of the slick to be exactly $4$ cm.

The volume of a sphere in terms 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 $1000$ cm$^3$ (take $\pi \approx 3.14$).

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

The intensity of light $I$ 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 $I(d) = k \times d^{-2}$. If at a distance of $2$ meters the intensity is $100$ lux, find the full expression for the function and calculate the intensity that the object will receive at a distance of $5$ meters.

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 $1200$ kg, express its kinetic energy as a power function dependent on its speed $v$ (in m/s) and calculate the energy it has when travelling at $20$ m/s.

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

The basal metabolic rate $T$ (in kcal/day) of mammals is modelled in terms of their body mass $M$ (in kg) using Kleiber's power law: $T(M) = 70 \times M^{3/4}$. Calculate the basal metabolic rate of a $16$ kg dog. Then, 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, $T^2 = k \times R^3$. If for Earth $T = 1$ 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 production cost of an electronic component decreases as the number of units manufactured increases according to the power function $C(x) = 100\,000 \times x^{-1/2}$, where $x$ is the quantity of components and $C(x)$ is the unit cost in pesos. 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 $500$ pesos.