Glossary
Closed Interval
An interval that includes its endpoints, denoted by square brackets [a, b], meaning both 'a' and 'b' are part of the interval.
Example:
For the IVT to apply, a function must be continuous on a closed interval like [0, 5], ensuring the behavior at the boundaries is considered.
Continuous Function
A function whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes over its domain or a specified interval.
Example:
The function f(x) = x² is a continuous function on all real numbers, making it suitable for applying theorems like IVT on any interval.
Intermediate Value
Any y-value that lies strictly between the function's values at the endpoints of a given interval, f(a) and f(b).
Example:
If f(1) = 3 and f(4) = 9, then 6 is an intermediate value that the continuous function f(x) must attain somewhere in the interval [1, 4].
Intermediate Value Theorem (IVT)
If a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b). It proves the existence of a value without finding it directly.
Example:
If a hiker starts at an altitude of 1000 ft and ends at 5000 ft on a continuous path, the Intermediate Value Theorem guarantees they passed through 3000 ft at some point.
Opposite Signs
When the function values at the endpoints of an interval, f(a) and f(b), have different signs (one positive and one negative). This condition, along with continuity, guarantees a root between 'a' and 'b' by IVT.
Example:
If f(0) = -3 and f(2) = 7, their opposite signs indicate that a continuous function must have a root somewhere between x=0 and x=2.
Root (of a function)
A value 'c' in the domain of a function where f(c) = 0, also commonly referred to as an x-intercept.
Example:
To find the root of f(x) = x - 5, we set f(x) = 0, which gives x = 5, meaning the graph crosses the x-axis at (5, 0).