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f(x)=x1,x=0
The graph of the function f(x)=x1 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=x1 can be used to graph other reciprocal functions. This can be done by applying different transformations.
Name | Equation | Characteristics |
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Parent Reciprocal Function | y=x1 | Domain:Range:Asymptotes: R−{0} R−{0} x- and y-axes
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Inverse Variation Functions | y=xa | |
General Form of Reciprocal Functions | y=x−ha+k | Domain:Range:Asymptotes: R−{h} R−{k} x=h and y=k
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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 a1, while the notation f -1(x) is commonly used to refer to the inverse of a function.
Function, f | Reciprocal, f1 | Inverse, f-1 |
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f(x)=2x+10 | f(x)1=2x+101 | f -1(x)=21(x−10) |
f(x)=x3−5 | f(x)1=x3−51 | f -1(x)=3x+5 |