This note covers odd and even functions, which have symmetric graphs, and inverse functions, including those that are their own inverse. These ideas let you describe a graph from its equation, undo a function, and understand when a function can be reversed.
Key ideas
Even functions. A function is even if \(f(-x)=f(x)\) for every \(x\) in its domain. Its graph is symmetric about the \(y\)-axis, so reflecting it in the \(y\)-axis leaves it unchanged. Examples are \(x^2\), \(x^4\), \(\cos x\) and \(|x|\).
Odd functions. A function is odd if \(f(-x)=-f(x)\) for every \(x\) in its domain. Its graph has rotational symmetry of order 2 about the origin: turning it through \(180^\circ\) about \((0,0)\) leaves it unchanged. Examples are \(x\), \(x^3\), \(\sin x\) and \(\dfrac1x\).
Most functions are neither odd nor even, such as \(x^2+x\) or \(2^x\). To test, find \(f(-x)\), simplify, and compare it with \(f(x)\) and with \(-f(x)\). If neither matches for all \(x\), the function is neither. A single counterexample is enough to show it is not odd or not even.
Inverse functions. The inverse \(f^{-1}\) reverses what \(f\) does. If \(f(a)=b\) then \(f^{-1}(b)=a\). For \(f^{-1}\) to exist as a function, \(f\) must be one-to-one: each output comes from exactly one input, which you can check with the horizontal line test. If \(f\) is not one-to-one, you can restrict its domain to make it so.
The inverse has these properties:
- \(f\big(f^{-1}(x)\big)=x\) and \(f^{-1}\big(f(x)\big)=x\).
- The domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\).
- The graph of \(y=f^{-1}(x)\) is the reflection of the graph of \(y=f(x)\) in the line \(y=x\).
Finding the inverse. Write \(y=f(x)\). Swap the roles of \(x\) and \(y\), or solve for \(x\) in terms of \(y\) and then rename. Rearrange to make \(y\) the subject. Then write the result as \(f^{-1}(x)\) and state its domain. When the function is a quadratic such as \(f(x)=(x-2)^2+1\) with \(x\ge2\), the restriction matters: solving \(y=(x-2)^2+1\) gives \(x=2+\sqrt{y-1}\) (positive root, because \(x\ge2\)), so \(f^{-1}(x)=2+\sqrt{x-1}\) for \(x\ge1\).
Self-inverse functions. A function is self-inverse if \(f^{-1}(x)=f(x)\), which is the same as \(f\big(f(x)\big)=x\). Its graph is symmetric about the line \(y=x\). Examples are \(f(x)=-x\), \(f(x)=\dfrac1x\) and, more generally, \(f(x)=\dfrac{ax+b}{cx-a}\), where the number subtracted in the denominator matches the coefficient of \(x\) in the numerator.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Even: \(f(-x)=f(x)\) | Graph symmetric about the \(y\)-axis. |
| Odd: \(f(-x)=-f(x)\) | Graph unchanged by a \(180^\circ\) turn about the origin. If \(f(0)\) exists, then \(f(0)=0\). |
| One-to-one function | Each output has exactly one input. Needed for an inverse to exist. |
| \(f\big(f^{-1}(x)\big)=x\) | Check that a proposed inverse is correct. |
| Domain of \(f^{-1}\) = range of \(f\) | Always state the domain of an inverse. |
| Graph of \(f^{-1}\) | Reflection of the graph of \(f\) in \(y=x\). |
| Self-inverse: \(f^{-1}=f\) | Equivalent to \(f(f(x))=x\). The graph is symmetric about \(y=x\). |
Worked examples
Example 1 Determine whether each function is odd, even or neither: (a) \(f(x)=x^3-4x\), (b) \(g(x)=x^2+\cos x\), (c) \(h(x)=x^2+x\).
(a) \(f(-x)=(-x)^3-4(-x)=-x^3+4x=-(x^3-4x)=-f(x)\). The function is odd.
(b) \(g(-x)=(-x)^2+\cos(-x)=x^2+\cos x=g(x)\), because cosine is even. The function is even.
(c) \(h(-x)=(-x)^2+(-x)=x^2-x\). This is not equal to \(h(x)=x^2+x\), and it is not equal to \(-h(x)=-x^2-x\) either. For a counterexample, \(h(1)=2\), \(h(-1)=0\), and neither \(0=2\) nor \(0=-2\) is true. The function is neither.
Answer: (a) odd, (b) even, (c) neither
Example 2 Find the inverse of \(f(x)=\dfrac{2x+1}{x-3}\) for \(x\ne3\). State the domain and range of \(f^{-1}\).
Write \(y=\dfrac{2x+1}{x-3}\). Multiply both sides by \((x-3)\): \(y(x-3)=2x+1\).
Expand: \(xy-3y=2x+1\).
Collect the terms in \(x\) on one side: \(xy-2x=3y+1\), so \(x(y-2)=3y+1\).
Divide: \(x=\dfrac{3y+1}{y-2}\). Rename \(y\) as \(x\) to give \(f^{-1}(x)=\dfrac{3x+1}{x-2}\).
The range of \(f\) is \(y\ne2\) (the horizontal asymptote). So the domain of \(f^{-1}\) is \(x\ne2\). The range of \(f^{-1}\) is the domain of \(f\), which is \(y\ne3\).
Check with a value: \(f(4)=\dfrac{9}{1}=9\), and \(f^{-1}(9)=\dfrac{28}{7}=4\).
Answer: \(f^{-1}(x)=\dfrac{3x+1}{x-2}\), domain \(x\ne2\), range \(y\ne3\)
Example 3 The function \(f(x)=\dfrac{kx+3}{x-2}\), \(x\ne2\), is self-inverse. Find the value of \(k\), and verify using \(f(f(x))=x\).
Find the inverse in terms of \(k\). Write \(y(x-2)=kx+3\), so \(xy-2y=kx+3\).
Then \(xy-kx=2y+3\), so \(x(y-k)=2y+3\), and \(f^{-1}(x)=\dfrac{2x+3}{x-k}\).
For \(f^{-1}=f\), compare \(\dfrac{2x+3}{x-k}\) with \(\dfrac{kx+3}{x-2}\). The denominators match when \(k=2\), and then the numerators also match: both are \(2x+3\).
Verify with \(k=2\): \(f(x)=\dfrac{2x+3}{x-2}\). Then \(f(f(x))=\dfrac{2\cdot\frac{2x+3}{x-2}+3}{\frac{2x+3}{x-2}-2}\).
Multiply the top and bottom by \((x-2)\): numerator \(=2(2x+3)+3(x-2)=7x\), denominator \(=(2x+3)-2(x-2)=7\).
So \(f(f(x))=\dfrac{7x}{7}=x\).
Answer: \(k=2\)
Common errors
- Confusing \(f^{-1}(x)\) with \(\dfrac{1}{f(x)}\). The inverse function is not the reciprocal.
- Testing evenness or oddness with a single value. A value like \(f(2)=f(-2)\) does not prove \(f\) is even. You need \(f(-x)\) for a general \(x\).
- Forgetting to state the domain of the inverse, or to restrict the domain of a function that is not one-to-one, such as \(x^2\).
- Taking both square roots when finding the inverse of a restricted quadratic. The domain of \(f\) decides whether the root is positive or negative.
- Assuming all functions are either odd or even. Many are neither.
Exam tip: Check your inverse by picking one input, applying \(f\), then your \(f^{-1}\). If you do not return to the start, find the algebra slip before moving on.
Check yourself
- State whether \(f(x)=x^5+x^3\) is odd, even or neither, and give a reason.
Answer
\(f(-x)=(-x)^5+(-x)^3=-x^5-x^3=-f(x)\), so \(f\) is odd. - Find the inverse of \(f(x)=3x-5\).
Answer
Let \(y=3x-5\). Then \(x=\dfrac{y+5}{3}\), so \(f^{-1}(x)=\dfrac{x+5}{3}\). - Find the inverse of \(f(x)=2e^x\) and state its domain.
Answer
Let \(y=2e^x\). Then \(e^x=\dfrac y2\), so \(x=\ln\dfrac y2\). Hence \(f^{-1}(x)=\ln\dfrac x2\). The range of \(f\) is \(y\gt0\), so the domain of \(f^{-1}\) is \(x\gt0\). - Show that \(f(x)=\dfrac{x+1}{x-1}\), \(x\ne1\), is self-inverse.
Answer
\(f(f(x))=\dfrac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}\). Multiplying top and bottom by \((x-1)\) gives \(\dfrac{(x+1)+(x-1)}{(x+1)-(x-1)}=\dfrac{2x}{2}=x\). So \(f=f^{-1}\).