Inverse function

A function f and its inverse  −1</sup>"}},"i":0}}]}'/>f. Because f maps 3 back to f (also called the inverse of f exists if and only if f:X→Y, its inverse x∈X such that f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the output, then divides the result by 5. Therefore, the inverse of f−1(y)=y+75.

X to Y back to X, and whose range is the set g from g(f(x))=x for all y∈Y.

If g is called the inverse of f is invertible if and only if it is injective. This is because the condition f is injective.

The inverse function  −1</sup>"}},"i":0}}]}'/>f to X and range x∈X and f−1∘f=idX and X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.

Considering function composition helps to understand the notation  −1</sup>"}},"i":0}}]}'/>f. Repeatedly composing a function n times, starting with the value f(x) = f (f (x)), etc. Since  −1</sup>(''f'' (''x'')) {{=}} ''x''"}},"i":0}}]}'/>f(f (x)) = x, composing  −1</sup>"}},"i":0}}]}'/>f and  ''n''</sup>"}},"i":0}}]}'/>f yields  ''n''−1</sup>"}},"i":0}}]}'/>f, "undoing" the effect of one application of (f(x)) certainly denotes the multiplicative inverse of f⟨−1⟩ might be used for the inverse function to avoid ambiguity with the multiplicative inverse.

In keeping with the general notation, some English authors use expressions like −1</sup>(''x'')"}},"i":0}}]}'/>sin(x) to denote the inverse of the sine function applied to (sin (x)). To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin ārea). For instance, the inverse of the hyperbolic sine function is typically written as sin−1(x)={(−1)narcsin⁡(x)+πn:n∈ℤ}. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the  −1</sup>"}},"i":0}}]}'/>f notation should be avoided.

The function (−x)2=x2 for all f:[0,∞)→[0,∞); x↦x2 with the same rule as before, then the function is bijective and so, invertible. The inverse function here is called the (positive) square root function and is denoted by