1YReciprocal

= Reciprocals =

Reciprocals of Fractions


 * The ** reciprocal  ** of any fraction is the result of flipping it upside down.

math . \qquad \text {So the reciprocal of } \dfrac {2}{5} \text { is } \dfrac {5}{2} math


 * More technically, the numerator and denominator have been swapped.

math . \qquad \text{The } \textbf{ reciprocal } \text{ of } n \text{ is } \dfrac{1}{n} math
 * Mathematically,

math . \qquad \text{The reciprocal of } \dfrac{2}{3} \text{ is } \dfrac{1}{\frac{2}{3}} = \dfrac{1 \times 3}{\frac{2}{3} \times 3} = \dfrac{3}{2} math


 * Whole numbers should be treated as having a denominator of 1

math \\ . \qquad 7 = \dfrac {7}{1} \\. \\ . \qquad \text {So the reciprocal of } 7 \text { is } \dfrac {1}{7} math


 * Notice that the __ product __ of a number and its reciprocal is always one.
 * this is called the ** mutiplicative inverse law  **

math . \qquad \text {Eg } \: \dfrac {2}{5} \times \dfrac {5}{2} = 1 math

math . \qquad \text {Eg } \: 7 \times \dfrac {1}{7} = 1 math


 * Notice that the reciprocal of 0 is __ **undefined** __.

Reciprocals of Algebraic Expressions


 * Reciprocals are often used with algebra as well

math . \qquad \text {For } x \text {, the reciprocal is } \dfrac {1}{x} math

math . \qquad \text {For } x^2 + 3 \text{, the reciprocal is } \dfrac {1}{x^2 + 3} math


 * Warning!! **

math \\ . \qquad \text{The reciprocal of } \; \dfrac{2}{3} + \dfrac{3}{4} \\. \\ . \qquad \textbf{IS NOT } \; \dfrac{3}{2} + \dfrac{4}{3} math


 * You have to first add the fractions together, then take the reciprocal:

math \\ . \qquad \dfrac{2}{3} + \dfrac{3}{4} = \dfrac{17}{12} \\. \\ . \qquad \text{so the reciprocal is } \dfrac{12}{17} math


 * The same is true with algebraic fractions

math \\ . \qquad \dfrac{2}{x} + \dfrac{3}{x+1} = \dfrac{2(x+1)}{x(x+1)} + \dfrac{3x}{x(x+1)} \\. \\ . \qquad \qquad \qquad = \dfrac{5x+2}{x^2+x} \\. \\ . \qquad \text{so the reciprocal is } \dfrac{x^2+x}{5x+2} math

Graphs of Reciprocal Functions


 * For any function, y = f(x), we can draw the graph of its reciprocal function:

math . \qquad y = \dfrac{1}{f(x)} math


 * For a demonstration of this process in Powerpoint, download the following file (5.4 Mbyte)
 * [[file:1A Reciprocal Graphs.ppsx]]


 * The technique is to consider different significant y-coordinates of the original graph and take the reciprocal of each y-value.


 * __All__ reciprocal graphs have a **horizontal asymptote** along the x-axis (y = 0)


 * For any x-intercepts (y = 0), the reciprocal will have a **vertical asymptote** at those points


 * Any points where y = 1 (or y = –1), the reciprocal is also 1 (or –1),
 * so the graph and its reciprocal will intercept at those points


 * When y is negative, the reciprocal is also negative


 * When |y| < 1, the reciprocal is |y| > 1
 * When |y| > 1, the reciprocal is |y| < 1


 * Pay attention to each end of the x-axis and close to any vertical asymptotes
 * As y+ → 0, y(reciprocal) → + ¥
 * so the reciprocal graph goes __up__ and approaches the __vertical asymptote__
 * As y → + ¥, y(reciprocal) → 0+
 * so the reciprocal graph approaches the __horizontal asymptote__ from __above__


 * Take care when drawing a graph close to an asymptote.
 * The graph should approach the asymptote, but not touch or curl away
 * The graph should approach the asymptote, but not touch or curl away


 * Example **

All of the points listed above can be seen occuring here:

We start with the line y = x – 1 (in blue)

Taking the reciprocal produces a hyperbola (in red):

math . \qquad y = \dfrac{1}{x-1} math


 * Domain: x Î R\{1}
 * Ramge: y Î R\{0}


 * Asymptotes:
 * x = 1
 * y = 0


 * y-intercept: (0, –1)

Worksheet

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