Another term equivalent to "the reciprocal function" is "the multiplicative inverse of the function."

Sometimes there is confusion between the terms "reciprocal" and "inverse", as "reciprocal" and "multiplicative inverse" mean the same thing, whereas "multiplicative inverse" and "inverse" may not necessarily refer to the same type of inverse.

Numbers (Addition Example)

The additive inverse of  4  is  -4.

The additive inverse of  -4  is  4.

4 + -4 = 0, and -4 + 4 = 0

Zero is the additive identity.

Numbers (Multiplication Example)

4 x 4^-1 = 1,  4^-1 x 4 = 1.

One is the multiplicative identity.

Functions (Addition Example)

The additive inverse of  f(x) = x - 2  is  -f(x) = -x + 2.

Note that  f(x) + -f(x) = 0  and  (x - 2) + (-x + 2) = 0

Note again that  zero  is the additive identity.

Functions (Multiplication Example)

One is the multiplicative identity.

NOTE: These statements are all true for permissible values in the domains of the function and corresponding reciprocal function.

Important

NOTE: Additive Inverse: f(x) + -f(x) = 0, the additive identity.

NOTE: Inverse:  f( f^-1(x) ) = x  and  f^-1( f(x) ) = x.

The following are some of the relationships between any function (and its graph) and the corresponding reciprocal function (and its graph).

• POSITIVE / NEGATIVE
  • If  f(x) > 0  then  [f(x)]^-1 > 0.
  • If  f(x) < 0  then  [f(x)]^-1 < 0.
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• RISING / FALLING
  • If  f(x)  is rising (left to right), then the reciprocal function is falling (left to right).
  • If  f(x)  is falling (left to right), then the reciprocal function is rising (left to right).
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• MAXIMA / MINIMA
  • Where  f(x)  has a non-zero maxima,  the reciprocal function has a non-zero minima.
  • Where  f(x)  has a non-zero minima,  the reciprocal function has a non-zero maxima.
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• INVARIANT POINTS
  • Where  f(x) = 1,  [f(x)]^-1 = 1.
  • Where  f(x) = -1,  [f(x)]^-1 = -1.
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• ASYMPTOTES AND LIMITS
  • Where  f(x) = 0,  the reciprocal function will have a vertical asymptote.
  • As  f(x)  increases towards zero, the reciprocal function decreases towards negative infinity.
  • As  f(x)  decreases towards zero, the reciprocal function increases towards positive infinity.
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• MORE LIMITS [ WHEN  f(x)  IS A POLYNOMIAL FUNCTION ]
  • As  f(x)  increases towards positive infinity, the reciprocal function decreases towards zero.
  • As  f(x)  decreases towards negative infinity, the reciprocal function increases towards zero.
  • Click here to see an example.