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Reciprocal Function — Definition, Formula & Examples

The reciprocal function is the function f(x)=1xf(x) = \frac{1}{x}, which outputs the multiplicative inverse of each input. Its graph forms a hyperbola with two branches, one in the first quadrant and one in the third quadrant.

The reciprocal function is defined as f:R∖{0}→R∖{0}f: \mathbb{R} \setminus \{0\} \to \mathbb{R} \setminus \{0\} by f(x)=1xf(x) = \frac{1}{x}. It has a vertical asymptote at x=0x = 0 and a horizontal asymptote at y=0y = 0, and it is an odd function satisfying f(−x)=−f(x)f(-x) = -f(x).

Key Formula

f(x)=1xf(x) = \frac{1}{x}
Where:
  • xx = Any real number except 0

How It Works

To evaluate the reciprocal function, simply divide 1 by the input value. For example, f(4)=14f(4) = \frac{1}{4} and f(−2)=−12f(-2) = -\frac{1}{2}. As xx gets closer to 0, the output grows without bound in magnitude, which creates the vertical asymptote. As ∣x∣|x| increases, the output approaches 0 but never reaches it, creating the horizontal asymptote. The function is decreasing on (−∞,0)(-\infty, 0) and also decreasing on (0,∞)(0, \infty), but it is not decreasing over its entire domain because of the discontinuity at x=0x = 0.

Worked Example

Problem: Find the output values of the reciprocal function for x=5x = 5, x=−13x = -\frac{1}{3}, and determine what happens as xx approaches 0 from the right.
Evaluate at x = 5: Substitute into the formula.
f(5)=15=0.2f(5) = \frac{1}{5} = 0.2
Evaluate at x = -1/3: Dividing 1 by a fraction flips it.
f ⁣(−13)=1−13=−3f\!\left(-\tfrac{1}{3}\right) = \frac{1}{-\frac{1}{3}} = -3
Behavior near 0: As x→0+x \to 0^+, the denominator shrinks toward 0 while the numerator stays 1, so the output grows without bound.
lim⁡x→0+1x=+∞\lim_{x \to 0^+} \frac{1}{x} = +\infty
Answer: f(5)=0.2f(5) = 0.2, f(−13)=−3f(-\frac{1}{3}) = -3, and f(x)→+∞f(x) \to +\infty as x→0+x \to 0^+.

Why It Matters

The reciprocal function is a parent function you will transform repeatedly in Algebra 2 and Precalculus when graphing rational functions. It also models real-world inverse relationships, such as the time to travel a fixed distance as speed varies.

Common Mistakes

Mistake: Assuming the reciprocal of xx is −x-x instead of 1x\frac{1}{x}.
Correction: The reciprocal (multiplicative inverse) of xx is 1x\frac{1}{x}, the value that multiplies with xx to give 1. The negative (additive inverse) of xx is −x-x. These are different operations.