Inverse function: Difference between revisions
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An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse. | An inverse function is a function that serves to "undo" another function. That is, if '''f(x)''' produces '''y''', then putting '''y''' into the inverse of '''f''' produces the output '''x'''. Not every function has an inverse. | ||
== Definition == | == Definition == | ||
A function g is the inverse function of f if | A function '''g''' is the inverse function of f if '''f(g(x))'''='''x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f"). | ||
=== Notation === | === Notation === | ||
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) | Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''. | ||
== Relevant observations == | == Relevant observations == | ||
* If g is the inverse function of f, then f is the inverse function of g. | * If '''g''' is the inverse function of '''f''', then '''f''' is the inverse function of '''g'''. | ||
* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f. | * The domain of '''f<sup>-1</sup>''' is the range of '''f''' and the range of '''f<sup>-1</sup>''' is the domain of '''f'''. | ||
* A function may not have an inverse function, but if it has, the inverse function is unique. | * A function may not have an inverse function, but if it has, the inverse function is unique. | ||
Latest revision as of 03:59, 28 April 2022
An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse.
Definition
A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f-1 ("inverse of f").
Notation
Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f-1(x)≠1/f(x).
Relevant observations
- If g is the inverse function of f, then f is the inverse function of g.
- The domain of f-1 is the range of f and the range of f-1 is the domain of f.
- A function may not have an inverse function, but if it has, the inverse function is unique.