Exercises: Parity of a function

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

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

Observe the following graph and classify the represented function as even, odd or with no obvious 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 $\$C(4) = 150$, 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 $p(3) = -12$ ft, what was its position at time $t=-3$ s?

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

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

Analytically determine the parity of the trigonometric function $f(x) = 2\sin(x)$, remembering the properties of the 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 to the center in feet. Prove analytically whether this profile is symmetric about the central vertical axis (even function).

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

The electromagnetic force of attraction between two metallic components is modeled by 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 have for this symmetry to be fulfilled.

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 satellite dish. Knowing that its global equation corresponds to an even function, describe what the left half would be 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 $x$.

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

A data analysis company uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetric error of a prediction algorithm. Analyze and prove the parity of this function.

Prove that any function $f(x)$ defined in $\mathbb{R}$ 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 $I(x) = \frac{f(x)-f(-x)}{2}$.

Applying the principle from the previous exercise, 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 in the interval $(-1, 1)$.

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

An engineer designs a mechanical part whose upper contour is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the contour 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 $c$.

Analyze 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 foci 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 complete analytical expression of $f(x)$.

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