Working with the Intermediate Value Theorem (IVT)
Use continuity and bracketing to justify an intermediate output, without claiming a unique solution.
College Board Topic 1.16 · CED p. 48 (PDF p. 53) · CED effective Fall 2020; project course outline 2026–27.
Continuity guarantees no skipped output values across an interval. The Intermediate Value Theorem says: if f is continuous on [a,b] and d lies between f(a) and f(b), there is a c in [a,b] with f(c)=d. If d is strictly between the endpoint values, c lies in (a,b).
For a root, choose d=0 and look for opposite-sign endpoint values. State the continuity hypothesis and compute the endpoint values before claiming existence. The theorem guarantees at least one such input; it does not locate it exactly or prove uniqueness.
For f(x)=x³+x−1 on [0,1], f is a polynomial, f(0)=−1 and f(1)=1. Zero is between those values, so at least one root lies in (0,1). A discontinuous function can jump from negative to positive without ever reaching zero, so endpoint signs alone are insufficient.
Let's work through it
- Let f(x)=x³+x−1. It is continuous on [0,1]. Reason: polynomials are continuous everywhere.
- Compute f(0)=−1 and f(1)=1; 0 lies strictly between them. Reason: these endpoint outputs bracket the desired value.
- Conclude a root c exists in (0,1). Reason: all hypotheses of the Intermediate Value Theorem hold.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Does IVT prove a root is unique?
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Retrieval 1
Does IVT prove a root is unique?
Worked answer and reason
No. It guarantees existence of at least one root.
Uniqueness needs additional information, such as strict monotonicity.
Retrieval 2
Why cannot IVT justify a root of 1/x between −1 and 1?
Worked answer and reason
The function is not continuous on [−1,1].
Its missing value and vertical asymptote at 0 violate the hypothesis.
Catch the tempting mistake
- Opposite endpoint signs always guarantee a zero.
- That conclusion also requires continuity throughout the closed interval.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 1 problem 28 · No calculator · verbal
- Unit 1 problem 29 · No calculator · verbal
- Unit 1 problem 30 · No calculator · graphical
For the teacher
- Use a jump function as a counterexample to signs-only reasoning.
- Grade the hypothesis, endpoint comparison and existence conclusion as separate reasoning steps.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.