Inverse function call the function turning initial dependence at = f(x) in such a way that an argument x and function at change roles. That is x becomes function from y (x = f (y)). At the same time schedules of mutually inverse functions at = to f (x) and x = are symmetric to f (y) in relation to ordinate axis in the first and third coordinate quarters of the Cartesian system. A range of definition of inverse function is the area of values initial, and area of values in turn – a range of definition of the set function.
Instruction
1. Generally when finding inverse function for f(x) set at = express an argument x through function at. For this purpose use rules of multiplication of both parts of equality by the same value, transfer of polynomials of expressions, at the same time consider sign change. In a simple case of consideration of exponential functions of a look: y = (7/x) + 11, the address of an argument x is made elementary: 7/x = at-11, x = 7 * (at-11). Required inverse function has an appearance x = 7 * (at-11).
2. However often in functions difficult sedate and logarithmic expressions and also trigonometrical functions are used. In this case when finding inverse function it is necessary to consider the known properties of these mathematical expressions.
3. If in initial function the argument x costs under degree, for receiving inverse function take from this expression a root with the same indicator. For example, for the set function at = 7+ x² the return will have an appearance: f (y) = u-7.
4. By function consideration where the argument x represents degree of constant number, apply definition of a logarithm. Follows from it that for the f (x) function = ah the return will be f (y) = logay, and the basis of a logarithm and – in both cases the number other than zero. Also and vice versa, considering the initial logarithmic f (x) function = logax, its inverse function represents sedate expression: f (y) = hey.
5. In that specific case researches of the function containing a natural logarithm of ln x or decimal lg x, i.e. logarithms on the number basis e and 10 respectively, receiving inverse function is carried out similarly, only instead of the basis and the exponential number or number 10 is substituted. For example, f (x) = lg x-> f (y) = 10u and f (x) = ln x-> f (y) = eu.
6. For trigonometrical functions by the return the following couples to each other are: - y = cos x-> x = arccos y; - y = sin x-> x = arcsin y; - y = tan x-> x = arctan y.