Glossary
Domain and Range Swap
A fundamental property of inverse functions where the domain of the original function becomes the range of its inverse, and the range of the original function becomes the domain of its inverse.
Example:
If f(x) = √x has a domain of [0, ∞) and a range of [0, ∞), then its inverse f⁻¹(x) = x² (for x ≥ 0) will have its Domain and Range Swap to a domain of [0, ∞) and a range of [0, ∞).
Horizontal Line Test
A graphical test used to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, the function is not one-to-one.
Example:
To check if f(x) = x² is one-to-one, apply the Horizontal Line Test; a horizontal line at y=4 would cross at x=2 and x=-2, so it fails the test.
Input-Output Pairs (for inverse functions)
The relationship between a function and its inverse where if (a, b) is an input-output pair for the original function, then (b, a) is the corresponding input-output pair for its inverse.
Example:
If f(3) = 7, then for the inverse function, the Input-Output Pairs dictate that f⁻¹(7) must equal 3.
Inverse Function Notation
The standard way to denote the inverse of a function f(x), which is written as f⁻¹(x).
Example:
If f(x) = x + 5, then its Inverse Function Notation would be f⁻¹(x) = x - 5.
Invertible Function
A function that has an inverse function, meaning it is both one-to-one and has an unrestricted domain, allowing its input-output relationship to be reversed.
Example:
The function f(x) = 2x + 1 is an invertible function because each output comes from a unique input, and its domain covers all real numbers.
Multivalued Inverses
Occurs when a function is not one-to-one, resulting in an inverse relation that is not a function because a single input in the inverse corresponds to multiple outputs.
Example:
The inverse of f(x) = x² is a Multivalued Inverse (y = ±√x) because for a single x-value (e.g., x=4), there are two y-values (y=2 and y=-2).
One-to-One Function
A function where each unique output (y-value) corresponds to only one unique input (x-value), ensuring no y-value is repeated for different x-values.
Example:
The function f(x) = x³ is a one-to-one function because every distinct x-value produces a distinct y-value.
Reflection (of inverse function graph)
The graphical relationship between a function and its inverse, where the graph of the inverse is a mirror image of the original function's graph across the line y = x.
Example:
When you graph f(x) = 2x and f⁻¹(x) = x/2, you'll see the graph of f⁻¹(x) is a perfect reflection of f(x) over the line y=x.
Swap Method (for finding inverse functions)
A step-by-step algebraic procedure to find the inverse of a function by replacing f(x) with y, swapping x and y in the equation, solving for y, and then replacing y with f⁻¹(x).
Example:
To find the inverse of f(x) = 4x - 1, use the Swap Method: y = 4x - 1 becomes x = 4y - 1, then solve for y to get y = (x + 1)/4.
Unrestricted Domain
A domain for a function that is not limited in a way that would prevent the function from being one-to-one, typically covering all real numbers or a continuous interval.
Example:
For f(x) = x³, its unrestricted domain of all real numbers allows it to be invertible without needing artificial limits.