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.

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 $C(-4)$?

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.

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

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).

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 $F_2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

Given the function $g(x) = x^3 \times \sin(x)$:\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.

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

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 $x$.

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 $f(x)$.

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

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 $I(x) = \frac{f(x)-f(-x)}{2}$.

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 $(-1, 1)$.

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.