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 ($f(-x) = -f(x)$).

In a 4th Year class, Cian must determine whether the function $f(x) = x^3 - 5x + 3$ is even, odd, or has no symmetry. Cian writes the following in his copy:

"Since the leading degree is 3 and the exponents of the variable $x$ are odd ($3$ and $1$), the function is odd. To check it, I calculate $f(-x) = (-x)^3 - 5(-x) + 3 = -x^3 + 5x + 3$, which, having different signs compared to the original formula, proves that it is odd".

1. a) Identify and explain the two conceptual errors that Cian has made (one when judging the function only by the exponents of $x$ forgetting the constant term, and another when interpreting what it means to change the sign of a function).
2. b) Use the value of the function at the origin, $f(0)$, to immediately justify why this function cannot be odd.
3. c) Correct the exercise by performing the appropriate algebraic verification to determine the actual symmetry of $f(x)$.

A function $g$, whose domain is all of $\mathbb{R}$, has odd symmetry and we only know three facts about it: $g(3) = -12$, $g(-1) = 4$, and that its graph intersects the vertical axis at the point $(0, k)$.

1. a) Reasonably deduce the numerical value of the constant $k$ and calculate the values of $g(-3)$ and $g(1)$ applying the algebraic definition of an odd function.
2. b) Write the analytic expression of a third-degree polynomial with odd symmetry, of the form $p(x) = a \times x^3 + b \times x$, that simultaneously satisfies $p(1) = -4$ and whose leading coefficient is $a = 2$.
3. c) Verify algebraically that the formula you obtained in section b rigorously satisfies the condition $p(-x) = -p(x)$ for any value of $x$.

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.

To find out whether the rational function $f(x) = \frac{3x^2 - 6}{x^4 + 2}$ is even, odd, or lacks symmetry, two students propose different approaches:

• Candela proposes the numerical method: "I calculate $f(2)$ and $f(-2)$. If both results match exactly, it is $100\,\%$ proven that the function is even".

• Hugo proposes the analytical method: "I calculate the general expression of $f(-x)$ by formally substituting $(-x)$ into the variable, expand, and compare it with $f(x)$ and $-f(x)$".

1. a) Evaluate the reliability of Candela's method: calculate $f(2)$ and $f(-2)$ and explain why verifying an equality for a single pair of opposite numbers does not constitute a valid mathematical proof for the entire domain.
2. b) Apply Hugo's method, clearly developing each algebraic step, and conclude with certainty what type of symmetry $f(x)$ exhibits.
3. c) Under what specific circumstance would a numerical method like Candela's actually be conclusive to resolve the symmetry problem? Justify your answer.

A Met Éireann station in County Galway records the temperature on an anticyclonic, windless winter day. Taking $14{:}00\text{ h}$ (the time of maximum solar radiation) as the reference time ($t = 0$), meteorologists model the temperature $T$ (in degrees Celsius) during the central daylight hours using the function:

$T(t) = -0.5 \times t^2 + 8$, for $t \in [-4, 4]$

(where $t$ indicates the hours elapsed relative to $14{:}00\text{ h}$; for example, $t = -3$ corresponds to $11{:}00\text{ h}$ and $t = 3$ corresponds to $17{:}00\text{ h}$).

1. a) Algebraically prove what symmetry the function $T(t)$ exhibits on the given interval by calculating $T(-t)$.
2. b) Interpret the physical meaning of this symmetry in the real meteorological context: what relationship is there between the temperatures at $11{:}00\text{ h}$ and at $17{:}00\text{ h}$? And at $12{:}00\text{ h}$ relative to $16{:}00\text{ h}$?
3. c) If a complementary report proposes modeling the rate of change of temperature using the function $v(t) = -1.0 \times t$, analyze its symmetry and interpret what the sign of $v(t)$ physically represents before $14{:}00\text{ h}$ ($t < 0$) compared to after $14{:}00\text{ h}$ ($t > 0$).

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.

Given two functions with domain on all of $\mathbb{R}$, $f(x) = x^3 - 4x$ (an odd function) and $g(x) = x^2 + 3$ (an even function), two new functions are defined from them: their product $P(x) = f(x) \times g(x)$ and their sum $S(x) = f(x) + g(x)$.

1. a) Reasonably estimate, before carrying out algebraic expansion calculations, what type of symmetry you would expect the product $P(x)$ to have (thinking about the sign rule for even and odd factors) and what symmetry you would expect the sum $S(x)$ to have when combining terms with different behavior under the opposite sign.
2. b) Check your estimates by algebraically calculating the expanded expressions of $P(-x)$ and $S(-x)$, and rigorously classify each of the two functions as even, odd, or without symmetry.
3. c) Reason through the following theoretical question: if we add any non-zero even function to any non-zero odd function, could the result ever be an even function? Justify your answer using the mathematical equality that defines even functions.

An industrial design team models the curved profile of a part using a third-degree polynomial function: $f(x) = a \times x^3 + b \times x^2 + c \times x + d$.

To fit into the mechanism, the part must meet three technical conditions:

1. Its graph must be strictly symmetric with respect to the origin of coordinates $(0,0)$.

2. The curve must intersect the horizontal axis at the point $(3, 0)$.

3. At the point with abscissa $x = 1$, the function must take the value $y = -8$.

1. a) Reasonably deduce the required values of the coefficients $b$ and $d$ without setting up systems of equations, relying exclusively on the analytical and geometric definition of odd symmetry.
2. b) Set up the system of linear equations corresponding to conditions 2 and 3, solve it, and write the final analytical formula for the function $f(x)$.
3. c) Prove that, for any odd function $f$ and for any point $x_0 \neq 0$ in its domain, the secant line connecting the point $A(x_0, f(x_0))$ with its symmetric point $B(-x_0, f(-x_0))$ necessarily passes through the origin of coordinates $(0,0)$.