Exercises: Symmetry of a function

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

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 $h(x) = x^3 - 4x + 1$.

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 $f(2) = -2$.

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.

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

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?

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.

In a 4th-year ESO exam, Hugo has to determine whether the function $f(x) = x^3 - 4x + 1$ is even, odd, or has no symmetry. Hugo writes on his paper:

"I calculate $f(-x) = (-x)^3 - 4(-x) + 1 = -x^3 + 4x + 1$. Since the terms with $x$ have all changed sign, the function is odd."

1. a) Find and explain in your own words the conceptual mistake that Hugo made when applying the definition of an odd function ($f(-x) = -f(x)$).
2. b) Write the full expression of $-f(x)$ and compare it term by term with $f(-x)$ to correct Hugo's reasoning.
3. c) Determine in a reasoned manner the correct conclusion regarding the symmetry of $f(x)$.

The Fuengirola town council is designing a new skate park and commissions the cross-sectional profile of a U-shaped ramp modeled by a quadratic function $f(x) = a \cdot x^2 + b \cdot x + c$, where $x$ is the horizontal distance in meters from the center of the track and $f(x)$ is the height in meters above the ground.

1. a) For safety and balance in the design, the ramp must have even symmetry with respect to the central vertical axis ($Y$). Prove algebraically what value the parameter $b$ must necessarily take so that $f(-x) = f(x)$ always holds.
2. b) Knowing that the lowest point of the ramp is at the center at ground level ($f(0) = 0$) and that $3\text{ m}$ from the center the height is $1,8\text{ m}$, find the values of $a$ and $c$, and write the final formula for the ramp.
3. c) Without making numerical substitutions in the formula, deduce what the value of $f(-3)$ will be, justifying it based on the symmetry of the 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 $a$.

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

Prove algebraically whether the combined exponential function $f(x) = \frac{3^x - 3^{-x}}{2}$ is even, odd, or neither.

The following graph shows only the part corresponding to $x \ge 0$ of a function $f(x)$. Given that the function is odd, describe what the missing part in the interval $x < 0$ would look like and determine the coordinates of the vertices on the negative side.

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.

The temperature variation in a test chamber is governed by the function $T(t) = \frac{12t}{t^2 + 4}$ for $t \in [-5, 5]$, where $t$ is the time in hours and $T$ is the temperature in degrees Celsius. Determine the symmetry of the function and interpret what happens to the temperatures at the times $t = 3$ and $t = -3$.

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 $g(x) = x$.

To find out whether the polynomial function $f(x) = 2x^4 - 7x^2 + 3$ has any type of symmetry, two students propose different strategies:

• Sofía proposes a numerical approach: calculate $f(2)$ and $f(-2)$, and if they give the same result, state that the function is even.

• Mateo proposes the analytical method: calculate the general algebraic expression of $f(-x)$ and compare it directly with $f(x)$ and with $-f(x)$.

1. a) Apply Mateo's method step by step and conclude with justification whether the function is even, odd, or has no symmetry.
2. b) Compare both methods and explain why Sofía's approach cannot be used to formally prove that a function is even across its entire domain.
3. c) However, Sofía's method is useful for ruling out symmetries. Check this with the function $g(x) = x^2 + 3x$: evaluate $g(1)$ and $g(-1)$, and justify from them why it can be immediately stated that $g$ is not even.

In the adaptation works of a road tunnel on the A-6 highway as it passes through Guadarrama, the roof height is modeled by the function $h(x) = 6 - 0,15 \cdot x^2$, where $x$ represents the horizontal distance in meters from the central dividing median (with $-5 \le x \le 5$) and $h(x)$ is the tunnel height in meters.

1. a) Before doing any calculations, estimate whether it makes engineering sense for the cross section of a two-way tunnel to be an even function, odd function, or without symmetry with respect to the center line, and explain your reasoning.
2. b) Verify algebraically whether the function $h(x)$ is even, odd, or lacks symmetry by applying the formal condition on $h(-x)$.
3. c) A fire truck travels in the right lane and its outermost point is located at $x = 4\text{ m}$ from the median. Calculate the height of the tunnel at that point to see if it clears a sign installed at a height of $3,5\text{ m}$. After that, justify using symmetry what the height will be in the left lane at $x = -4\text{ m}$ without repeating the calculation.

Find the values of the parameters $a$ and $b$ such that the rational function $f(x) = \frac{ax^3 + b}{x^2 - 1}$ is odd and passes through the point $(2, \frac{16}{3})$.

Prove that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function and an odd function. (Hint: consider the expressions $\frac{f(x)+f(-x)}{2}$ and $\frac{f(x)-f(-x)}{2}$).

An architectural arch is modelled using the function $y = \frac{k}{e^{cx} + e^{-cx}}$. Prove that this arch is symmetrical about the y-axis regardless of the value of $c$. If the maximum height of the arch at $x = 0$ is $10$ metres, find the value of $k$.

Determine algebraically the type of symmetry of the logarithmic function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$. Take into account the properties of logarithms.

Find the values of the parameters $m$ and $n$ that make the polynomial function $f(x) = x^4 + (m-2)x^3 + nx^2 + (n+1)x + 5$ symmetrical about the y-axis.

The magnetic field $B(x)$ created by an infinitely long straight conducting wire at a distance $x$ is given by $B(x) = \frac{\mu_0 I}{2\pi x}$. Check the parity of this function. What physical interpretation does it have that the function is odd when evaluating the field at a position $x$ and at its opposite $-x$?

If $f(x)$ is an even function and $g(x)$ is any function, determine with reasoning the symmetry of the composite function $k(x) = g(f(x))$. Can the same be said for the function $h(x) = f(g(x))$?

An even function $f(x)$ is formed by line segments. In the first quadrant, it encloses a triangle with vertices $(0, 4)$, $(3, 0)$ and the origin. From the graph shown for $x \ge 0$, deduce the equation of $f(x)$ across its entire domain expressed as a piecewise function and calculate the total area enclosed by the function and the x-axis.

The Spanish Institute of Oceanography (IEO) monitors water flow at the mouth of a Galician ría. The speed of the water in meters per second during a cycle is modeled by the function $v(t) = \frac{4t}{t^2 + 4}$, where $t$ is the time in hours elapsed from the moment of mean water level ($t = 0$), with $-3 \le t \le 3$. Positive values indicate incoming current (high tide) and negative values indicate outgoing current (low tide).

1. a) Prove analytically that the function $v(t)$ is odd by calculating $v(-t)$ step by step and comparing it with $-v(t)$.
2. b) If 2 hours after mean tide ($t = 2$) the sensor records an inflow speed of $1\text{ m/s}$, deduce with reasoning based on symmetry what the speed and direction of the water were 2 hours before said central instant ($t = -2$).
3. c) A team member states that any odd function whose domain includes $0$ must necessarily pass through the origin $(0,0)$. Prove algebraically whether this statement is always true using the equality $f(-x) = -f(x)$ for $x = 0$, and explain what physical meaning this has for the current at $t = 0$.

The State Meteorological Agency (AEMET) models atmospheric pressure around the eye of a storm in the Cantabrian Sea using $P(x) = 0,04x^2 + 990$, where $x$ is the distance in tens of kilometers to the east ($x > 0$) or to the west ($x < 0$) of the storm's center ($x = 0$), and $P(x)$ is the pressure in hectopascals ($\text{hPa}$).

1. a) Justify analytically why $P(x)$ is an even function and what meteorological implication this has regarding the pressure distribution on both sides of the storm's eye.
2. b) A front displaces the storm $30\text{ km}$ to the east (that is, 3 tens of kilometers to the right), so that the new model becomes $Q(x) = P(x - 3) = 0,04(x - 3)^2 + 990$. Expand the expression for $Q(x)$ and check algebraically whether $Q(x)$ remains an even function with respect to the $Y$-axis.
3. c) Explain from a geometric point of view why a horizontal shift causes the loss of even symmetry with respect to the $Y$-axis, state what the new vertical axis of symmetry of the graph is now, and reason whether a vertical translation (for example, adding $5\text{ hPa}$ to the entire function) would also cause the loss of even symmetry.