Even and Odd Functions
Picture the cross-section of a skate park half-pipe. The left wall is a perfect mirror image of the right wall. Now picture a Ferris wheel: spin it 180° and it looks exactly the same. Some functions behave like the half-pipe, some like the Ferris wheel, and most like neither. Can you tell which is which from a formula alone, without drawing anything?
Even, odd, and symmetry
Two kinds of symmetry show up again and again in graphs. One is a mirror image across the y-axis. The other is a 180° rotation about the origin.
Why the names? Powers of $x$ decide it. $x^2, x^4, x^6$ are even functions, and $x, x^3, x^5$ are odd functions.
To read parity from a graph, look at matching pairs of points at $x = a$ and $x = -a$:
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Same height at $x = a$ and $x = -a$ means even.
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Opposite heights (same distance from the x-axis, one above and one below) means odd.
If you can see only the right half of a graph, the parity tells you how to finish it. For an even function, reflect the right half across the y-axis. For an odd function, rotate it 180° about the origin.
Observe the following graph and classify the represented function as even, odd or with no obvious symmetry, justifying your answer based on the highlighted points.
Observe the following graph and classify the represented function as even, odd or with no obvious symmetry, justifying your answer based on the highlighted points.
The graph shows the right half (for $x \ge 0$) of the profile of a satellite dish. Knowing that its global equation corresponds to an even function, describe what the…
The graph shows the right half (for $x \ge 0$) of the profile of a satellite dish. Knowing that its global equation corresponds to an even function, describe what the left half would be like and what value $f(-x)$ will take as a function of $f(x)$.
Using parity to find values
Parity is a shortcut. If you know $f(a)$ and you know the function is even or odd, you instantly know $f(-a)$.
Why does it work? The definition $f(-x) = \pm f(x)$ holds for every $x$, so you can plug in any number, such as $x = a$.
Pick any $x$ on the graph above, such as $x = 1$, and then look at $x = -1$. The heights are opposites, because this function is odd.
Complete the missing values so that the function $f(x)$ is odd in its entire domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$
Complete the missing values so that the function $f(x)$ is odd in its entire domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$
Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $\$C(4) = 150…
Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $\$C(4) = 150$, what will be the value of $C(-4)$?
The position of an object in a three-dimensional virtual environment is governed by an odd function $p(t)$. If at time $t=3$ s its position on one of the axes is…
The position of an object in a three-dimensional virtual environment is governed by an odd function $p(t)$. If at time $t=3$ s its position on one of the axes is $p(3) = -12$ ft, what was its position at time $t=-3$ s?
Testing parity with a formula
Graphs and points can mislead, so to be certain you work with the formula.
For polynomials there is a fast check. A power $x^n$ becomes $(-x)^n$, which equals $x^n$ when $n$ is even and $-x^n$ when $n$ is odd. A constant counts as $x^0$, which is even. So:
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All terms have even powers (including a constant): the polynomial is even.
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All terms have odd powers: the polynomial is odd.
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A mix of even and odd powers: the polynomial is neither.
The first curve is symmetric about the y-axis. Adding the odd term $x$ destroys the symmetry in the second.
Analytically determine whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd or neither.
Analytically determine whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd or neither.
Prove mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ has any type of symmetry about the y-axis or about the origin.
Prove mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ has any type of symmetry about the y-axis or about the origin.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, indicating whether it has symmetry about the y-axis or about the origin.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, indicating whether it has symmetry about the y-axis or about the origin.
Trig, absolute value, and exponentials
The $f(-x)$ test works for any formula. A few basic functions are worth memorizing.
Why are sine and cosine different? On the unit circle, the angle $-x$ is the reflection of the angle $x$ across the horizontal axis. The reflected point keeps the same $x$-coordinate and flips the sign of the $y$-coordinate.
So $\cos(-x) = \cos x$ (even) and $\sin(-x) = -\sin x$ (odd). Dividing gives $\tan(-x) = \dfrac{-\sin x}{\cos x} = -\tan x$, so tangent is odd.
For a fraction, "opposite" means changing the sign of the whole fraction once, as in $-f(x) = \dfrac{-P(x)}{Q(x)}$.
Analytically determine the parity of the trigonometric function $f(x) = 2\sin(x)$, remembering the properties of the sine for negative angles.
Analytically determine the parity of the trigonometric function $f(x) = 2\sin(x)$, remembering the properties of the sine for negative angles.
The profile of an architectural arch is modeled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance to the center in feet. Prove…
The profile of an architectural arch is modeled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance to the center in feet. Prove analytically whether this profile is symmetric about the central vertical axis (even function).
A data analysis company uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetric error of a prediction algorithm. Analyze and prove the parity of this function.
A data analysis company uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetric error of a prediction algorithm. Analyze and prove the parity of this function.
Products and quotients
You can often skip the full computation by splitting a function into pieces whose parity you already know.
Why does it work? For a product $h = f \cdot g$ you get $h(-x) = f(-x)\,g(-x)$. Each odd factor pulls out one minus sign. Two odd factors give $(-1)(-1) = 1$. One odd factor gives a single $-1$.
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \cdot g(x)$ has.
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \cdot g(x)$ has.
Study the parity of the function $f(x) = x|x|$, which describes the aerodynamic drag force of a vehicle as a function of its speed $x$.
Study the parity of the function $f(x) = x|x|$, which describes the aerodynamic drag force of a vehicle as a function of its speed $x$.
Analytically determine whether the function $g(x) = \frac{x^3 - x}{|x|}$ is even, odd or neither.
Analytically determine whether the function $g(x) = \frac{x^3 - x}{|x|}$ is even, odd or neither.
Parity of a composition
When one function is plugged into another, $h(x) = f(g(x))$, parity also follows rules.
Why does it work? Always start with the inner function: $h(-x) = f(g(-x))$. If $g$ is even, $g(-x) = g(x)$, so the minus sign vanishes before $f$ ever sees it.
If $f(x)$ is an even function and $g(x)$ is an odd function, rigorously determine and prove the parity of the composite function $h(x) = f(g(x))$.
If $f(x)$ is an even function and $g(x)$ is an odd function, rigorously determine and prove the parity of the composite function $h(x) = f(g(x))$.
Finding unknown parameters
Sometimes a formula has a missing number, and a symmetry fixes it. The idea is to write the condition $f(-x) = f(x)$ (or $-f(x)$) and make it true for every $x$.
Here is a shortcut for polynomials. For an even polynomial, every odd-power coefficient must be $0$. For an odd polynomial, every even-power coefficient (including the constant) must be $0$.
Drag $a$ until the parabola is symmetric about the y-axis. That happens at $a = 0$, where the odd-power term $ax$ disappears.
Calculate the value of the parameter $k$ so that a company's profit function, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetric about…
Calculate the value of the parameter $k$ so that a company's profit function, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetric about $x=0$.
The electromagnetic force of attraction between two metallic components is modeled by an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the…
The electromagnetic force of attraction between two metallic components is modeled by an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the distance. Determine the value that the parameter $b$ must necessarily have for this symmetry to be fulfilled.
Let $f(x)$ be a polynomial function of degree 4. If $f(x)$ is an even function, $f(0)=3$, $f(1)=4$ and $f(2)=11$, determine the complete analytical expression of $f(x)$.
Let $f(x)$ be a polynomial function of degree 4. If $f(x)$ is an even function, $f(0)=3$, $f(1)=4$ and $f(2)=11$, determine the complete analytical expression of $f(x)$.
Trickier cases
Some functions hide their parity until you rewrite $f(-x)$ in a clever way. Look for these patterns.
Logarithm of a fraction. Replacing $x$ with $-x$ often flips the fraction upside down. Then use $\ln\!\left(\dfrac{1}{u}\right) = -\ln u$.
Two pieces that swap. If $-x$ makes two terms trade places, as in $A - B$ becoming $B - A$, then $f(-x) = -f(x)$.
Study the parity of the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$, used in physics to model certain electric potentials in the interval $(-1, 1)$.
Study the parity of the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$, used in physics to model certain electric potentials in the interval $(-1, 1)$.
Analyze the parity of the function $f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1}$, which represents the difference in distances from a point $P(x, 0)$ on the x-axis to…
Analyze the parity of the function $f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1}$, which represents the difference in distances from a point $P(x, 0)$ on the x-axis to two heat-emitting foci located at $F_1\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $F_2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
An engineer designs a mechanical part whose upper contour is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the contour is an odd function, passes through the…
An engineer designs a mechanical part whose upper contour is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the contour is an odd function, passes through the points $(2, \frac{10}{3})$ and $(3, \frac{15}{4})$, and has a vertical asymptote at $x=1$ (for $x>0$), calculate the exact values of the parameters $a, b$ and $c$.
Even part plus odd part
Most functions are neither even nor odd, but every function defined for all real numbers can be split into an even piece plus an odd piece.
Why? Add the two fractions and the $f(-x)$ terms cancel, leaving $\dfrac{2f(x)}{2} = f(x)$. Replace $x$ with $-x$ in $P$ and the two terms just swap, so $P$ is even. In $I$ the swap flips the numerator's sign, so $I$ is odd.
For polynomials you don't need the formula: the even part is the terms with even powers, and the odd part is the terms with odd powers.
Prove that any function $f(x)$ defined in $\mathbb{R}$ can be expressed as the sum of an even function $P(x)$ and an odd function $I(x)$ using the expressions…
Prove that any function $f(x)$ defined in $\mathbb{R}$ can be expressed as the sum of an even function $P(x)$ and an odd function $I(x)$ using the expressions $P(x) = \frac{f(x)+f(-x)}{2}$ and $I(x) = \frac{f(x)-f(-x)}{2}$.
Applying the principle from the previous exercise, decompose the exponential function $f(x) = e^x$ into the sum of an even function and an odd function.
Applying the principle from the previous exercise, decompose the exponential function $f(x) = e^x$ into the sum of an even function and an odd function.
Parity and derivatives
If you have seen derivatives, parity gives a nice connection. The derivative $f'(x)$ is the slope of the tangent line, and a symmetry of the graph forces a symmetry of the slopes.
The odd function $x^3$ has the even derivative $3x^2$. Look at the tangent to $x^3$ at $x = 1$ and at $x = -1$. Rotating the graph by 180° keeps the tangent lines parallel, so the slopes are equal.
The derivative of a function gives us the slope of its tangent line at each point. If $f(x)$ is an odd function and differentiable in all of $\mathbb{R}$, prove…
The derivative of a function gives us the slope of its tangent line at each point. If $f(x)$ is an odd function and differentiable in all of $\mathbb{R}$, prove mathematically that its derivative function $f'(x)$ must necessarily be an even function.
Back to the half-pipe: its profile can be modeled by a function like $h(x) = 0.05x^2$, and $h(-x) = h(x)$ because only an even power appears. That is exactly what "left wall is a mirror image of the right wall" means. A Ferris wheel's rotational symmetry is the odd case, with $f(-x) = -f(x)$.