<p class="paragraph-header">Even and Odd Functions</p>
<p>A function \(f\) is even if&nbsp;\(f(-x)=f(x)\) for all \(x\) in its domain.</p>
<p>A function \(f\) is odd if&nbsp;\(f(-x)=-f(x)\) for all \(x\) in its domain.</p>
<p>A function is neither even nor odd if&nbsp;\(f(-x)\ne f(x)\) and \(f(-x)\ne -f(x)\).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Type of function</th><th style="text-align: left; --cell-justify: flex-start">Algebraic condition</th><th style="text-align: left; --cell-justify: flex-start">Graphical meaning</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Even</td><td style="text-align: left; --cell-justify: flex-start">\(f(-x)=f(x)\)</td><td style="text-align: left; --cell-justify: flex-start">Symmetrical about the \(y\)-axis</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Odd</td><td style="text-align: left; --cell-justify: flex-start">\(f(-x)=-f(x)\)</td><td style="text-align: left; --cell-justify: flex-start">Rotational symmetry of \(180^\circ\) about the origin</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Neither</td><td style="text-align: left; --cell-justify: flex-start">\(f(-x)\ne f(x)\) and \(f(-x)\ne -f(x)\)</td><td style="text-align: left; --cell-justify: flex-start">No required symmetry</td></tr></tbody></table></div></div>
<p>To test whether a function is even, odd, or neither, first find \(f(-x)\). Then compare it with \(f(x)\) and \(-f(x)\).</p>
<ul class="bullet-list"><li data-list-item-id="even-odd-notes-7a" data-list-item-path="even-odd-notes-7a"><div class="list-item-paragraph">If&nbsp;\(f(-x)=f(x)\), then&nbsp;\(f\) is even.</div></li><li data-list-item-id="even-odd-notes-7b" data-list-item-path="even-odd-notes-7b"><div class="list-item-paragraph">If&nbsp;\(f(-x)=-f(x)\), then&nbsp;\(f\) is odd.</div></li><li data-list-item-id="even-odd-notes-7c" data-list-item-path="even-odd-notes-7c"><div class="list-item-paragraph">If neither condition is true, then&nbsp;\(f\) is neither even nor odd.</div></li></ul>
<p>Important IB HL point: the domain must be symmetrical about \(0\). For example, \([-3,3]\) is symmetrical about \(0\), but \([0,3]\) is not.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Function</th><th style="text-align: left; --cell-justify: flex-start">Type</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(x^2\)</td><td style="text-align: left; --cell-justify: flex-start">Even</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(x^4-3x^2+1\)</td><td style="text-align: left; --cell-justify: flex-start">Even</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(\cos x\)</td><td style="text-align: left; --cell-justify: flex-start">Even</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(x^3\)</td><td style="text-align: left; --cell-justify: flex-start">Odd</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(x^5-2x\)</td><td style="text-align: left; --cell-justify: flex-start">Odd</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(\sin x\)</td><td style="text-align: left; --cell-justify: flex-start">Odd</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(x^2+x\)</td><td style="text-align: left; --cell-justify: flex-start">Neither</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(x^3+1\)</td><td style="text-align: left; --cell-justify: flex-start">Neither</td></tr></tbody></table></div></div>
<p>Useful IB HL calculus results:</p>
<ul class="bullet-list"><li data-list-item-id="even-odd-notes-11a" data-list-item-path="even-odd-notes-11a"><div class="list-item-paragraph">If&nbsp;\(f\) is even, then&nbsp;\(\int_{-a}^{a} f(x)\,dx=2\int_0^a f(x)\,dx\).</div></li><li data-list-item-id="even-odd-notes-11b" data-list-item-path="even-odd-notes-11b"><div class="list-item-paragraph">If&nbsp;\(f\) is odd, then&nbsp;\(\int_{-a}^{a} f(x)\,dx=0\).</div></li></ul>
<p>Common mistake: a function with odd powers plus a constant is usually neither odd nor even. For example, \(f(x)=x^3+1\) is neither even nor odd.</p>

A function \(f\) is even if \(f(-x)=f(x)\) for all \(x\) in its domain.

A function \(f\) is odd if \(f(-x)=-f(x)\) for all \(x\) in its domain.

A function is neither even nor odd if \(f(-x)\ne f(x)\) and \(f(-x)\ne -f(x)\).

To test whether a function is even, odd, or neither, first find \(f(-x)\). Then compare it with \(f(x)\) and \(-f(x)\).

If neither condition is true, then \(f\) is neither even nor odd.

Important IB HL point: the domain must be symmetrical about \(0\). For example, \([-3,3]\) is symmetrical about \(0\), but \([0,3]\) is not.

If \(f\) is even, then \(\int_{-a}^{a} f(x)\,dx=2\int_0^a f(x)\,dx\).

If \(f\) is odd, then \(\int_{-a}^{a} f(x)\,dx=0\).

Common mistake: a function with odd powers plus a constant is usually neither odd nor even. For example, \(f(x)=x^3+1\) is neither even nor odd.

Odd and Even Functions

Functions

International Baccalaureate Higher Level Mathematics

Functions - Odd and Even Functions

Functions - Odd and Even Functions