Exercises: Parity of a function

Determine analytically whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd, or neither.

Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, stating whether it exhibits symmetry about the y-axis or about the origin.

Observe the following graph and classify the function shown as even, odd, or without evident symmetry, justifying your answer based on the highlighted points.

Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $C(4) = 150$ euros, what will be the value of $C(-4)$?

The position of an object in a three-dimensional virtual environment is governed by an odd function $p(t)$. If at time $t=3$ s its position on one of the axes is $p(3) = -12$ m, what was its position at time $t=-3$ s?

Check mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ exhibits any type of symmetry about the y-axis or about the origin.

Complete the missing values so that the function $f(x)$ is odd throughout its domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$

Determine analytically the parity of the trigonometric function $f(x) = 2\sin(x)$, recalling the properties of sine for negative angles.

Lucía claims in class: «Since in the function $f(x) = 2x^3 - 5x + 3$ the exponents of the $x$'s are $3$ and $1$ (both odd), the function must be odd. To prove it, I calculate $f(-x) = 2(-x)^3 - 5(-x) + 3 = -2x^3 + 5x + 3 = -(2x^3 - 5x - 3)$, and since almost all the terms change sign, it can be considered odd except for the constant».\na) Identify the conceptual and algebraic error made by Lucía when applying the definition of an odd function.\nb) Analytically determine whether $f(x)$ is even, odd, or neither.\nc) Reason what minimal modification should be made to the constant term so that the resulting function is indeed symmetric with respect to the origin.

Given the function $g(x) = x^3 \cdot \sin(x)$:\na) Estimate its parity before calculating: analyse the symmetry of each factor separately ($x^3$ and $\sin(x)$) and, by applying the sign rule for the product of symmetric functions, anticipate whether you expect $g(x)$ to be even, odd, or neither.\nb) Formally verify your estimate by calculating $g(-x)$ step by step and using the trigonometric property of $\sin(-x)$.\nc) Numerically evaluate $g\left(\frac{\pi}{2}\right)$ and $g\left(-\frac{\pi}{2}\right)$. Do these values match what the obtained parity predicts? Justify your answer.

The profile of an architectural arch is modelled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the centre in metres. Prove analytically whether this profile is symmetrical about the vertical central axis (even function).

Calculate the value of the parameter $k$ so that the profit function of a company, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetrical about $x=0$.

The electromagnetic attraction force between two metal components is modelled using an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the distance. Determine the value that the parameter $b$ must necessarily take for this symmetry to hold.

Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \cdot g(x)$ has.

The graph shows the right half (for $x \ge 0$) of the profile of a parabolic antenna. Knowing that its overall equation corresponds to an even function, describe what the left half would look like and what value $f(-x)$ will take as a function of $f(x)$.

Study the parity of the function $f(x) = x|x|$, which describes the aerodynamic drag force of a vehicle as a function of its speed $x$.

Determine analytically whether the function $g(x) = \frac{x^3 - x}{|x|}$ is even, odd, or neither.

A data analytics firm uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetrical error of a prediction algorithm. Analyse and prove the parity of this function.

At an AEMET monitoring station in an area of high solar radiation, the thermal oscillation with respect to the daily mean temperature during a clear equinox day is approximated by the model $T(t) = \frac{12 \cdot \cos(0,26 \cdot t)}{1 + 0,01 \cdot t^2}$, where $t$ is the time in hours elapsed since solar noon ($t = 0$, which corresponds to 13:30 official time), defined for the interval $t \in [-12, 12]$.\na) Analytically demonstrate whether the temperature model $T(t)$ is an even function, an odd function, or exhibits no symmetry.\nb) Physically interpret the result: what does this model predict about the temperature at 10:30 official time compared to the temperature at 16:30 official time?\nc) If, starting in the evening hours, the usual wind introduces an additional linear cooling modelled by a term $-0,4 \cdot t$ across the entire expression, reason whether the new resulting function would maintain symmetry with respect to solar noon.

Consider the family of rational functions $f(x) = \frac{a \cdot x^3 + (b - 4) \cdot x^2 + 6x}{x^2 + 2}$, where $a$ and $b$ are real parameters.\na) Find the values of $a$ and $b$ such that the function $f(x)$ is odd and its graph passes through the point $(1, 3)$.\nb) Is it possible to choose values for $a$ and $b$ such that $f(x)$ is an even function? If you think it is possible, calculate them; if it is not, justify algebraically why no pair of real values can achieve it.

Prove that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x)$ and an odd function $I(x)$ using the expressions $P(x) = \frac{f(x)+f(-x)}{2}$ and $I(x) = \frac{f(x)-f(-x)}{2}$.

Knowing that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x) = \frac{f(x)+f(-x)}{2}$ and an odd function $I(x) = \frac{f(x)-f(-x)}{2}$, decompose the exponential function $f(x) = e^x$ into the sum of an even function and an odd function.

Study the parity of the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$, used in physics to model certain electric potentials on the interval $(-1, 1)$.

If $f(x)$ is an even function and $g(x)$ is an odd function, determine and rigorously prove the parity of the composite function $h(x) = f(g(x))$.

An engineer designs a mechanical part whose upper outline is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the outline is an odd function, passes through the points $(2, \frac{10}{3})$ and $(3, \frac{15}{4})$, and has a vertical asymptote at $x=1$ (for $x>0$), calculate the exact values of the parameters $a, b$ and $c$.

Analyse the parity of the function $f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1}$, which represents the difference in distances from a point $P(x, 0)$ on the x-axis to two heat emitting sources located at $F_1\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $F_2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Let $f(x)$ be a polynomial function of degree 4. If $f(x)$ is an even function, $f(0)=3$, $f(1)=4$ and $f(2)=11$, determine the full analytical expression of $f(x)$.

The derivative of a function gives us the gradient of its tangent line at each point. If $f(x)$ is an odd and differentiable function on all of $\mathbb{R}$, prove mathematically that its derivative function $f'(x)$ must necessarily be an even function.

To study the parity of the function $h(x) = \frac{x^3 \cdot \sin(x^2)}{1 + \cos(x)}$, two different approaches are proposed:\n- Method 1 (Direct evaluation): Systematically substitute $-x$ for $x$ throughout the entire expression, simplify by applying the parity trigonometric identities, and compare $h(-x)$ with $h(x)$ and $-h(x)$.\n- Method 2 (Algebra of symmetries): Decompose $h(x)$ into the product and quotient of the elementary blocks $f_1(x) = x^3$, $f_2(x) = \sin(x^2)$, and $f_3(x) = 1 + \cos(x)$, analyse the parity of each block using the rules of composition and addition, and deduce the overall symmetry.\na) Apply both methods detailing all analytical steps.\nb) Compare both approaches: which one turns out to be safer and more efficient to avoid sign errors in complex nested functions? Justify your stance.\nc) Using the method that you justified as more efficient in the previous section, directly deduce the parity of the modified function $k(x) = \frac{x^3 \cdot \sin^2(x)}{1 + \cos(x)}$ without carrying out the complete direct calculation.

Design the algebraic expression of a rational function $f(x) = \frac{P(x)}{Q(x)}$ (where $P(x)$ and $Q(x)$ are polynomials with real coefficients, having no common factors, and of the lowest possible degree) that simultaneously satisfies these four conditions:\n1. Its domain is exactly $\mathbb{R} \setminus \{-2, 2\}$.\n2. $f(x)$ is an odd function throughout its domain.\n3. Its graph intersects the x-axis at the point $x = 1$.\n4. The graph passes through the point $(3; -4,8)$.\n\nTo construct it, answer the following steps with reasoning:\na) Deduce the polynomial of the denominator $Q(x)$ and justify what parity it has.\nb) Demonstrate what mandatory parity the numerator $P(x)$ must have for the quotient to satisfy the condition of being odd.\nc) Deduce all the mandatory zeros that $P(x)$ must have based on symmetry and write the factorised expression of $P(x)$ in terms of a constant $a$.\nd) Determine the exact value of the constant $a$ using the given point and write the final simplified formula for $f(x)$.