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f(x) = 1/x, x≠0
The graph of the function f(x) = 1x is a hyperbola, which consists of two symmetrical parts called branches. It has two asymptotes, the x- and y-axes. The domain and range are all nonzero real numbers.
The graph of y= 1x can be used to graph other reciprocal functions. This can be done by applying different transformations.
| Name | Equation | Characteristics |
|---|---|---|
| Parent Reciprocal Function | y=1/x | Domain: & R-{0} Range: & R-{0} Asymptotes: & x- andy-axes |
| Inverse Variation Functions | y=a/x | |
| General Form of Reciprocal Functions | y=a/x-h+k | Domain: & R-{h} Range: & R-{k} Asymptotes: & x = h andy=k |
The general form of reciprocal functions is also known as the simple rational function.
It is important not to confuse the reciprocal of a function with the inverse of a function. For numbers, a^(-1) refers to the reciprocal 1a, while the notation f^(- 1)(x) is commonly used to refer to the inverse of a function.
| Function, f | Reciprocal, 1f | Inverse, f^(-1) |
|---|---|---|
| f(x) = 2x + 10 | 1/f(x)=1/2x+10 | f^(- 1)(x) = 1/2(x - 10 ) |
| f(x) = x^3 - 5 | 1/f(x)=1/x^3-5 | f^(- 1)(x) = sqrt(x + 5) |