Symmetry of Functions (Even and Odd)
Picture the main cable of a suspension bridge like the Golden Gate Bridge. The curve on the left of the center looks exactly like the curve on the right. If you knew one half, you would get the other half for free. Which functions have this "free half" property, and how can you prove it with algebra instead of just eyeballing a picture?
Symmetry on a graph
Two kinds of symmetry show up again and again in graphs of functions. Each one has a physical test.
The mirror test: fold the page along the $y$-axis. If the left half lands exactly on the right half, the graph is symmetric about the $y$-axis.
The half-turn test: spin the page $180^\circ$ around the origin. If the graph lands exactly on itself, the graph is symmetric about the origin.
In the left figure the two marked points are mirror images: same height, opposite $x$-values. In the right figure the two marked points are opposite through the origin: both coordinates change sign.
Look at the graphs of the functions shown in panels A and B. State what type of symmetry (even, odd, or neither) each of them has, justifying your answer based on their…
Look at the graphs of the functions shown in panels A and B. State what type of symmetry (even, odd, or neither) each of them has, justifying your answer based on their visual shape.
Even functions: mirror in the $y$-axis
A picture can fool you, so we turn the mirror test into an algebra test. Reflecting the point $(x, f(x))$ in the $y$-axis gives $(-x, f(x))$. For the graph to land on itself, that point must also be on the graph, which means $f(-x) = f(x)$.
Why do powers decide so much? Because of the sign rule: $(-x)^n = x^n$ when $n$ is even, and $(-x)^n = -x^n$ when $n$ is odd. So even powers ignore the sign of $x$. The absolute value does too: $|-x| = |x|$.
Play with the slider below. When $h = 0$ the parabola is even. Slide $h$ away from $0$ and watch the mirror property disappear.
Given the polynomial function $f(x) = 3x^2 - 5$, calculate the value of $f(-x)$. From the result, state whether the function has even, odd, or no symmetry.
Given the polynomial function $f(x) = 3x^2 - 5$, calculate the value of $f(-x)$. From the result, state whether the function has even, odd, or no symmetry.
Consider the absolute value function $f(x) = |x| - 3$. Calculate the expression for $f(-x)$ and determine whether the function has any type of symmetry.
Consider the absolute value function $f(x) = |x| - 3$. Calculate the expression for $f(-x)$ and determine whether the function has any type of symmetry.
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, odd, or…
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, odd, or no symmetry, and interpret the result geometrically.
Odd functions: half-turn about the origin
Spinning the point $(x, f(x))$ by $180^\circ$ about the origin gives $(-x, -f(x))$. For the graph to land on itself, that point must be on the graph too. That means $f(-x) = -f(x)$.
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?
Analyze the linear function $f(x) = 4x$. Calculate $f(-x)$ and determine algebraically whether it is an even function, an odd function, or if it has no symmetry.
Analyze the linear function $f(x) = 4x$. Calculate $f(-x)$ and determine algebraically whether it is an even function, an odd function, or if it has no symmetry.
Given the polynomial function $f(x) = x^3 - 4x$, calculate the numerical values of $f(2)$ and $f(-2)$. What do these results suggest about the symmetry of the function?…
Given the polynomial function $f(x) = x^3 - 4x$, calculate the numerical values of $f(2)$ and $f(-2)$. What do these results suggest about the symmetry of the function? Prove your hypothesis algebraically for any value of $x$.
Functions with no symmetry
Most functions are neither even nor odd. Mixing even and odd powers is usually enough, and so is a nonzero constant added to odd powers.
Consider the linear function $f(x) = 2x + 1$. Check algebraically whether it satisfies the condition for even symmetry ($f(-x) = f(x)$) or odd symmetry ($f(-x) = -f(x)$).
Consider the linear function $f(x) = 2x + 1$. Check algebraically whether it satisfies the condition for even symmetry ($f(-x) = f(x)$) or odd symmetry ($f(-x) = -f(x)$).
Using symmetry to find values and points
Once you know a function is even or odd, one known value gives you a second one for free.
This is also why we can study only half the domain. The right half ($x \ge 0$) forces the left half: a mirror image if $f$ is even, a half-turn if $f$ is odd.
It is known that a function $h(x)$ has even symmetry. If the graph of this function passes through the point $(4, 7)$, what other point must necessarily belong to its…
It is known that a function $h(x)$ has even symmetry. If the graph of this function passes through the point $(4, 7)$, what other point must necessarily belong to its graph?
A function $p(x)$ has odd symmetry. If we know that $p(2) = -5$, what must the value of $p(-2)$ be? Write the coordinates of the corresponding point.
A function $p(x)$ has odd symmetry. If we know that $p(2) = -5$, what must the value of $p(-2)$ be? Write the coordinates of the corresponding point.
A company logo has been designed using a mathematical function that is symmetrical about the origin. If the outline of the logo passes through the point $(-3, 6)$, what…
A company logo has been designed using a mathematical function that is symmetrical about the origin. If the outline of the logo passes through the point $(-3, 6)$, what other point will it definitely pass through?
Odd functions at $x = 0$, and symmetric models
Odd functions have a special fact at $x = 0$.
This gives a fast way to rule out oddness. The function $f(x) = \dfrac{x^3 + 1}{x^2 + 1}$ is defined at $0$ and $f(0) = 1 \ne 0$, so it cannot be odd. (A value of $0$ at $x = 0$ does not prove oddness, though. It is only a necessary condition.)
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 this using the definition of an…
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 this using the definition of an odd function.
The path of a pendulum as a function of time $t$ is modeled by a function $p(t)$. Knowing that $p(t)$ is an odd function and that at $1.5$ seconds its position is…
The path of a pendulum as a function of time $t$ is modeled by a function $p(t)$. Knowing that $p(t)$ is an odd function and that at $1.5$ seconds its position is $p(1.5) = 4$ inches, what will its position be at $t = -1.5$ seconds? Explain the physical meaning of this result.
The following graph shows the temperature variation $T$ in a chemical experiment as a function of time $t$, recorded from the instant $t = 0$. If the mathematical model…
The following graph shows the temperature variation $T$ in a chemical experiment as a function of time $t$, recorded from the instant $t = 0$. If the mathematical model $T(t)$ governing this experiment is an even function over its entire real domain, what will the temperature be at the instant $t = -3$? Justify your answer.
Can a function be both even and odd?
Look at the two definitions side by side:
$$f(-x) = f(x) \qquad \text{and} \qquad f(-x) = -f(x).$$Usually a function satisfies one, or neither. But what if a function satisfied both? Treat it like a system of equations, with the two unknown quantities $f(x)$ and $f(-x)$.
Suppose there is a function $f(x)$ that is simultaneously even and odd over its entire 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 over its entire domain. Set up the system of equations derived from the definitions of both symmetries and solve it to find the only function that satisfies this condition.
Finding an unknown parameter
Sometimes a function contains an unknown constant, and the question is: for which value is it even, or odd? The strategy is to compute $f(-x)$, set it equal to $f(x)$ (even) or $-f(x)$ (odd), and compare the terms.
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.
Determine the value of the parameter $m$ for the polynomial function $f(x) = 2x^3 + m x^2 - x$ to have odd symmetry.
Determine the value of the parameter $m$ for the polynomial function $f(x) = 2x^3 + m x^2 - x$ to have odd symmetry.
Calculate the value of the parameter $b$ for the rational function $f(x) = \frac{5x^2 + bx - 2}{x}$ to be an odd function.
Calculate the value of the parameter $b$ for the rational function $f(x) = \frac{5x^2 + bx - 2}{x}$ to be an odd function.
Rational functions
For a rational function, substitute $-x$ into the numerator and the denominator separately, and watch the sign each power produces.
The curve passes through the origin, and the left piece is the right piece turned half a revolution.
Check algebraically the type of symmetry that the rational function $f(x) = \frac{5x}{x^2 + 2}$ has.
Check algebraically the type of symmetry that the rational function $f(x) = \frac{5x}{x^2 + 2}$ has.
Determine whether the function $g(x) = \frac{x^2 - 4}{x^2 + 1}$ is even, odd, or neither. Show all the steps of your reasoning.
Determine whether the function $g(x) = \frac{x^2 - 4}{x^2 + 1}$ is even, odd, or neither. Show all the steps of your reasoning.
The power generated by a wind turbine is modeled by the function $P(v) = \frac{v^3}{v^2 + 10}$, where $v$ represents the wind speed. Study analytically the symmetry of…
The power generated by a wind turbine is modeled by the function $P(v) = \frac{v^3}{v^2 + 10}$, where $v$ represents the wind speed. Study analytically the symmetry of $P(v)$. If we consider negative wind speeds as wind blowing in the opposite direction, what physical meaning does the obtained symmetry have with respect to the generated power?
Domains, and products of functions
The domain must be symmetric too
Symmetry compares $x$ and $-x$, so both must be in the domain. If $x = 2$ is allowed but $x = -2$ is not, the test already fails.
Products of even and odd functions
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 claim 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 claim correct? Prove or refute it using algebraic analysis.
To predict a financial trend, a model is created by multiplying two simpler functions: $h(x) = f(x) \cdot g(x)$. Knowing that $f(x) = x^2$ (an even function) and…
To predict a financial trend, a model is created by multiplying two simpler functions: $h(x) = f(x) \cdot g(x)$. Knowing that $f(x) = x^2$ (an even function) and $g(x) = x$ (an odd function), determine algebraically what type of symmetry the resulting function $h(x)$ will have.
Rigorously analyze the symmetry of the rational function $f(x) = \frac{x^3 - x}{x^2 - 4}$. Show all the algebraic steps.
Rigorously analyze the symmetry of the rational function $f(x) = \frac{x^3 - x}{x^2 - 4}$. Show all the algebraic steps.
Back to the bridge: a cable shaped like $y = ax^2 + c$ is even, since only even powers appear. That is exactly why the left half is a mirror image of the right half, and why engineers only need to design one half.