Parity of a Function

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

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.

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 $C(4) = 3\,000$ pesos, 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?

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

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

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 modeled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the center in meters. Prove analytically whether this profile is symmetrical about the vertical central axis (even function).

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 $x$.

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.

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 $(-1, 1)$.

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

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.

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 $c$.

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 $f(x)$.

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.

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

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.