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[h] IB Mathematics AA HL Flashcards- The rational function
[q] The rational function
This is the function in the form: \( f(x) = \frac{1}{x} \), and it has certain properties. Firstly, \( x \neq 0 \) because of division by zero. Also, \( y \neq 0 \) because \( \frac{1}{x} \) never equals zero. This corresponds to a horizontal and vertical asymptote at \( y = 0 \) & \( x = 0 \). Plugging in some values reveals its distinctive shape.
[q] SELF-INVERSE:
You may notice that it is symmetrical in \( y = x \), and if you find its inverse algebraically, \( f^{-1}(x) \) also equals \( \frac{1}{x} \). This tells us that \( f(x) = \frac{1}{x} \) is a self-inverse. This is also true for all \( f(x) = \frac{k}{x} \).
[a] RATIONAL FUNCTIONS:
This is the more general form of \( f(x) = \frac{ax + b}{cx + d} \), which has a similar shape: (shown graphs of different shapes). It is mostly just defined by the location of its asymptotes:
[q] Vertical Asymptotes:
This involves finding an \( x \) value that cannot be used. Which, in turn, means finding when the denominator \( = 0 \), i.e. \( cx + d = 0 \), \( x = -\frac{d}{c} \). (Marked V.A.)
[a] Horizontal Asymptotes:
This involves finding a value that the fraction \( \frac{ax + b}{cx + d} \) will never equal, but get infinitely close to. Plugging in a very large \( x \) shows that \( b \) and \( d \) become negligible, and \( f(x) \) ‘tends to’ \( \frac{a}{c} \), or just \( y = \frac{a}{c} \). (Marked H.A.)
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