Exercises: 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 outputs for $x = -2$, $x = 0$ and $x = 3$.

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

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 $(2, 40)$.

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

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.

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 $n$.

Lucía is analyzing the properties of two power functions, $f(x) = 3 \cdot x^{-2}$ and $g(x) = 2 \cdot x^{1/4}$, and writes in her notebook:\n1. "Since $f(0) = 3 \cdot 0^{-2} = 0$, the graph of $f$ intersects the $Y$-axis at the origin $(0, 0)$".\n2. "For the function $g$, since the fourth root of $16$ is $2$, if we input a negative value we get $g(-16) = 2 \cdot (-16)^{1/4} = 2 \cdot (-2) = -4$, so its domain includes all real numbers".\nIdentify with reasoning the conceptual error made in each of the two statements, explain why her calculation is incorrect, and state the correct real domain of each of these functions.

Write the algebraic expression of a power function of the form $f(x) = a \cdot x^n$ that simultaneously satisfies the following conditions:\na) Its exponent $n$ is a negative integer and its graph is symmetric with respect to the $Y$-axis (even function, where $f(-x) = f(x)$).\nb) It has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.\nc) Its graph passes through the point $P(2; 0,75)$.\nJustify the choice of the simplest possible value of $n$ that verifies the conditions and calculate in detail the value of the coefficient $a$, expressing it as an irreducible fraction or decimal number.

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 modelled by the power function $I(d) = k \cdot d^{-2}$. If at a distance of $2$ metres 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 $5$ metres.

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

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.

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.

The radius $r$ (in cm) of a circular oil slick spreading on water grows with time $t$ (in minutes) according to the power function $r(t) = 2 \cdot t^{1/2}$. 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.

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 $(2, 40)$.

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).

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 $\pi \approx 3,14$).

In a wind farm in Aragón, the electrical power $P$ (in kilowatts, kW) produced by a wind turbine is modeled by the power function $P(v) = k \cdot v^3$, where $v$ represents the wind speed (in m/s). Telemetry data indicates that with a wind speed of $5\text{ m/s}$, the power generated is $50\text{ kW}$.\na) Before performing any algebraic calculation, make a reasoned estimate: if the wind speed doubles to $10\text{ m/s}$, will the power double ($100\text{ kW}$), quadruple ($200\text{ kW}$), or increase even more? Justify your conjecture using the properties of the cubic exponent.\nb) Find the exact value of the constant $k$ from the initial data.\nc) Calculate analytically the power $P(10)$ and check whether your estimate was correct by calculating the exact ratio between $P(10)$ and $P(5)$.

A power function of the form $f(x) = a \cdot x^n$ passes through the points $A(2, 6)$ and $B(6, 54)$. To determine the exact values of the parameters $a$ and $n$, two classmates propose different approaches:\n- Javier's method (Direct quotient): He sets up the system $\begin{cases} 6 = a \cdot 2^n \\ 54 = a \cdot 6^n \end{cases}$, divides the second equation by the first to cancel the coefficient $a$, and groups the powers with the same base: $\frac{54}{6} = \left(\frac{6}{2}\right)^n \implies 9 = 3^n$.\n- Elena's method (Logarithmic linearization): She takes common logarithms on both sides of each equation to transform the system into a linear system of two equations with two unknowns, $\log(a)$ and $n$, then solving it by elimination.\na) Complete the steps of Javier's method and calculate the values of $n$ and $a$.\nb) Compare both procedures and justify which of the two is more efficient in this particular case.\nc) Explain what feature the coordinates of the points would need to have so that Javier's method could not be solved immediately at a glance and resorting to Elena's method would be justified.

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: $T(M) = 70 \cdot M^{3/4}$. 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.

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 behaviour of each one.

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

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 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 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 $10$ 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.

Find the algebraic expression for 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.

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 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 euros. 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 $25$ euros.

According to technical studies by the Directorate-General for Traffic (DGT), the braking distance $d$ (in meters) of a passenger car on dry asphalt until coming to a complete stop is modeled by a power function $d(v) = a \cdot v^n$, where $v$ is the driving speed (in km/h). In type-approval tests, it is observed that at $40\text{ km/h}$ the braking distance is $8\text{ meters}$, while at $80\text{ km/h}$ it rises to $32\text{ meters}$.\na) Analytically determine the expression of the function by deducing the integer exponent $n$ and the constant $a$.\nb) In Spain, the speed limit on single-lane urban roads in each direction was reduced from $50\text{ km/h}$ to $30\text{ km/h}$. Calculate the braking distance associated with each speed and determine the exact percentage reduction that this measure represents.\nc) A driver claims: "If I drive at $140\text{ km/h}$ on a highway instead of the $120\text{ km/h}$ limit, since I am only going $16,7\,\%$ faster, my braking distance will barely increase by that same percentage". Mathematically refute their argument based on the behavior of the power function and calculate the actual percentage increase in their braking distance.

You want to design the formula for a power function $f(x) = a \cdot x^n$ that models the residual concentration of a substance and simultaneously meets four mathematical conditions:

1. Its domain does not include negative values or zero, being naturally restricted to $(0, +\infty)$.

2. It has the Cartesian axes as asymptotes: a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.

3. If we multiply the value of the independent variable by $9$, the value returned by the function is reduced to exactly one third; that is, $f(9x) = \frac{1}{3} \cdot f(x)$ for all $x > 0$.

4. The graph passes through the point $P(4; 2,5)$.

1. a) From conditions 1 and 2, deduce with reasoning whether the exponent $n$ must be positive or negative, and why it must be a fraction with an even denominator.

1. b) Apply condition 3 to find the exact value of the exponent $n$.

1. c) Determine the value of the constant $a$ using the coordinates of point $P$ and write the final function both in power form and in radical form.