Locating roots by change of sign
Objective. Use interval evaluations to justify the location of a root.
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- 2. Worked example
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Learn: the key idea
Locating roots by change of sign uses the fact that if a continuous function f(x) is negative at one value and positive at another, a root of f(x)=0 must lie somewhere between them, since the graph must cross the x-axis to change sign. To justify a root lies in an interval [a,b], calculate f(a) and f(b) and show they have opposite signs. Narrowing the interval by testing values in between allows the root to be pinpointed to any required accuracy. For example, if f(2)=−3 and f(3)=5, a root of f(x)=0 lies between x=2 and x=3 because the sign changes from negative to positive.
Worked example
Show that the equation x³−x−3=0 has a root between x=1 and x=2.
- 1Define f(x) = x³−x−3
- 2Evaluate f(1) = 1−1−3 = −3, which is negative
- 3Evaluate f(2) = 8−2−3 = 3, which is positive
- 4Since f(1) is negative and f(2) is positive, the function changes sign, so a root lies between 1 and 2
Answer: A root lies in (1, 2) since f(1)=−3 and f(2)=3 differ in sign
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Start practisingCommon mistake. Students sometimes conclude there is no root simply because f(a) and f(b) are both large numbers, without checking their signs; what matters is whether the sign changes between the two values, not their size.
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