Symmetry of a function

Determine algebraically whether the function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd or has no symmetry.

Observe the following graph of a function $f(x)$. From its representation, determine what type of symmetry it possesses and calculate the value of $f(-2)$ knowing that $f(2) = -2$.

Check analytically whether the rational function $g(x) = \frac{2x}{x^2 + 1}$ shows any type of symmetry.

Study the symmetry of the polynomial function $h(x) = x^3 - 4x + 1$.

Classify the following elementary functions as even, odd or having no symmetry: $y = x$, $y = x^2$, $y = \frac{1}{x}$, $y = |x|$.

The graph below represents the function $g(x)$. Visually identify its symmetry and find $g(-3)$ if we know that $g(3) = 2$.

You can do it on paper and send a photo with your answer.

A function $f(x)$ passes through the point $(2, 5)$. If we know that it is an even function, through which other point must it pass? What if it were an odd function?

The main cable of a suspension bridge takes the form of a parabola described by the function $f(x) = 0,05x^2 + 10$, where $x$ is the horizontal distance in metres from the centre of the bridge. Test its symmetry algebraically and explain the physical meaning of this result for the structure of the bridge.

The profile of a wind turbine blade is modelled using an odd function of the form $f(x) = ax^3$. If the tip of the blade is located at the coordinates $(15, 30)$ in cm, at what coordinates will the opposite tip be located? Calculate also the value of the parameter $a$.

Reason whether the product of an even function and an odd function results in an even function, an odd function, or a function with no symmetry. You can use simple examples such as $x^2$ and $x$ to illustrate your conclusion.

Study the symmetry of the function $f(x) = \frac{x^3}{\sqrt{x^2+4}}$ algebraically.

Find the value that the parameter $k$ must take for the function $f(x) = 2x^2 + kx + 5$ to be an even function across its entire domain.

Given two functions, $f(x)$ (even) and $g(x)$ (odd), analyse the symmetry of the sum function $h(x) = f(x) + g(x)$. Verify this using the functions $f(x) = x^2$ and $g(x) = x$.