Symmetry of a function
Before sketching a function it is worth asking whether it has any symmetry: it saves half the work and, in Bachillerato, it is the first thing you analyse when studying any function. Symmetry is not decided by looking at a drawing, but by proving it with an algebraic substitution.
Even symmetry: reflection in 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: point symmetry about the origin
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 for them to mix even and odd exponents, or to have a non-zero constant term alongside terms with odd exponents.
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.
Practical uses of knowing the symmetry
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.
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 calculated by substituting separately in the numerator and the denominator, paying attention to the sign introduced by 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.