Even and Odd Functions (Symmetry)

Look at the graphs of the functions shown in panels A and B. State what type of symmetry (even, odd, or neither) each of them has, justifying your answer based on their visual shape.

Given the polynomial function $f(x) = 3x^2 - 5$, calculate the value of $f(-x)$. From the result, state whether the function has even, odd, or no symmetry.

Consider the absolute value function $f(x) = |x| - 3$. Calculate the expression for $f(-x)$ and determine whether the function has any type of symmetry.

Given the function $g(x) = -2x^3$, calculate the expression for $g(-x)$. What type of symmetry does this function have?

Analyze the linear function $f(x) = 4x$. Calculate $f(-x)$ and determine algebraically whether it is an even function, an odd function, or if it has no symmetry.

Given the polynomial function $f(x) = x^3 - 4x$, calculate the numerical values of $f(2)$ and $f(-2)$. What do these results suggest about the symmetry of the function? Prove your hypothesis algebraically for any value of $x$.

Consider the linear function $f(x) = 2x + 1$. Check algebraically whether it satisfies the condition for even symmetry ($f(-x) = f(x)$) or odd symmetry ($f(-x) = -f(x)$).

A student claims that the function $f(x) = \frac{1}{x-2}$ is odd because the $x$ in the denominator is raised to an odd exponent (1). Is this claim correct? Prove or refute it using algebraic analysis.

Rigorously analyze the symmetry of the rational function $f(x) = \frac{x^3 - x}{x^2 - 4}$. Show all the algebraic steps.

It is known that a function $h(x)$ has even symmetry. If the graph of this function passes through the point $(4, 7)$, what other point must necessarily belong to its graph?

A function $p(x)$ has odd symmetry. If we know that $p(2) = -5$, what must the value of $p(-2)$ be? Write the coordinates of the corresponding point.

A company logo has been designed using a mathematical function that is symmetrical about the origin. If the outline of the logo passes through the point $(-3, 6)$, what other point will it definitely pass through?

The path of a pendulum as a function of time $t$ is modeled by a function $p(t)$. Knowing that $p(t)$ is an odd function and that at $1.5$ seconds its position is $p(1.5) = 4$ inches, what will its position be at $t = -1.5$ seconds? Explain the physical meaning of this result.

The cable of a suspension bridge takes the shape of a parabola described by the equation $y = 0.05x^2 + 10$. Verify algebraically whether this function has even, odd, or no symmetry, and interpret the result geometrically.

The following graph shows the temperature variation $T$ in a chemical experiment as a function of time $t$, recorded from the instant $t = 0$. If the mathematical model $T(t)$ governing this experiment is an even function over its entire real domain, what will the temperature be at the instant $t = -3$? Justify your answer.

Check algebraically the type of symmetry that the rational function $f(x) = \frac{5x}{x^2 + 2}$ has.

Determine whether the function $g(x) = \frac{x^2 - 4}{x^2 + 1}$ is even, odd, or neither. Show all the steps of your reasoning.

The power generated by a wind turbine is modeled by the function $P(v) = \frac{v^3}{v^2 + 10}$, where $v$ represents the wind speed. Study analytically the symmetry of $P(v)$. If we consider negative wind speeds as wind blowing in the opposite direction, what physical meaning does the obtained symmetry have with respect to the generated power?

Knowing that $f(x)$ is an odd function and that it is defined at the point $x = 0$, what must the value of $f(0)$ necessarily be? Prove this using the definition of an odd function.

To predict a financial trend, a model is created by multiplying two simpler functions: $h(x) = f(x) \cdot g(x)$. Knowing that $f(x) = x^2$ (an even function) and $g(x) = x$ (an odd function), determine algebraically what type of symmetry the resulting function $h(x)$ will have.

Suppose there is a function $f(x)$ that is simultaneously even and odd over its entire domain. Set up the system of equations derived from the definitions of both symmetries and solve it to find the only function that satisfies this condition.

Find the value that the parameter $k$ must take for the function $f(x) = x^2 + kx + 5$ to be an even function. Justify your answer algebraically.

Determine the value of the parameter $m$ for the polynomial function $f(x) = 2x^3 + m x^2 - x$ to have odd symmetry.

Calculate the value of the parameter $b$ for the rational function $f(x) = \frac{5x^2 + bx - 2}{x}$ to be an odd function.