Exercises: Symmetry

Look at the graphs of the functions represented in panels A and B. State what type of symmetry (even, odd or none) each of them displays, 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 symmetry, odd symmetry or neither.

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

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

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

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 symmetry, odd symmetry or neither, and interpret the result geometrically.

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

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 it using the definition of an odd function.

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.

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

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 statement correct? Prove or refute it using an algebraic analysis.

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