Symmetry of a Function
Ximena records the temperature of an experiment starting at $t = 0$ and wonders whether she can guess what would have happened at negative times without measuring anything. If the function has symmetry, she can. And symmetry is not decided "by eye": it is proven with an algebraic substitution.
Even symmetry: reflection across the $Y$-axis
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 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.
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 (…
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 ($f(-x) = -f(x)$).
Odd symmetry: about the origin
Can a function be even and odd at the same time?
If $f$ satisfies both definitions, then $f(-x) = f(x)$ and $f(-x) = -f(x)$. Setting them equal, $f(x) = -f(x)$, that is, $2f(x) = 0$, and therefore $f(x) = 0$ for every $x$ in the domain. The only function that is both even and odd is the constant zero function.
Given the function $g(x) = -2x^3$, calculate the expression for $g(-x)$. What type of symmetry does this function have?
Given the function $g(x) = -2x^3$, calculate the expression for $g(-x)$. What type of symmetry does this function have?
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…
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.
Functions with no symmetry
Most functions are neither even nor odd: it is enough to mix even and odd exponents, or to have a nonzero constant term together with terms of odd exponent.
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…
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.
What knowing the symmetry is good for
Back to the beginning: if Ximena's temperature model is even, the temperature at $t = -3$ is the same as at $t = 3$.
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…
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.
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…
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.
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.
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.
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…
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.
Symmetry in rational functions
In a rational function, $f(-x)$ is computed by substituting in both the numerator and the denominator, paying attention to the sign of each odd power.
Check algebraically the type of symmetry shown by the rational function $f(x) = \frac{5x}{x^2 + 2}$.
Check algebraically the type of symmetry shown by the rational function $f(x) = \frac{5x}{x^2 + 2}$.
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 $P(v)$…
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…
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.