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Switching from one form of equation to another is common when working with rational functions.
A rational function is a function of the form |P/Q| where both the numerator and the denominator are polynomials.
Here, we are interested when the polynomials |P| and |Q| are linear functions.
||f(x)= \frac{P}{Q} = \frac{ax+b}{cx+d}|| Note: When the two polynomials are linear functions, this form of the rule is called the general form or homographic form of a rational function.
It is necessary for at least one of the two parameters |a| and |c| to be non-zero.
Here is how to convert the standard form |\displaystyle \frac{a}{b(x-h)}+k| to the general form |\displaystyle \frac{ax+b}{cx+d}|.
Put the two fractions over the same denominator.
Add the two numerators.
||\begin{align} f(x) &= \dfrac{3}{2(x+1)}-2 \\ &= \dfrac{3}{2(x+1)}-\dfrac{2\times \color{#ec0000}{2(x+1)}}{\color{#ec0000}{2(x+1)}} \\ &= \dfrac{3}{2(x+1)} - \dfrac{4(x+1)}{2(x+1)} \\ &= \dfrac{3-4(x+1)}{2(x+1)} \\ &= \dfrac{3-4x-4}{2(x+1)} \\ &= \dfrac{-4x-1}{2x+2} \end{align}||
Here is how to convert from the general form |\displaystyle \frac{ax+b}{cx+d}| to the standard form |\displaystyle \frac{a}{b(x-h)}+k|.
Divide the numerator by the denominator.
Transform the following rational function. ||f(x) = \dfrac{-4x-1}{2x+2}||
First, divide the numerator by the denominator. ||\require{enclose} \begin{array}{rll} -2 \\[-3pt] 2x+2 \enclose{longdiv}{-4x-1}\kern-.2ex \\[-3pt] \underline{-(-4x-4)}\\[-3pt] 3 \\[-3pt] \end{array}||
The |\color{#3a9a38}{3}| corresponds to the remainder, |\color{#ec0000}{2x+2}| remains the denominator, and |\color{#333fb1}{-2}| is the integer result of the division, that is, the parameter |k.|
So, the result of the division is the following. ||\begin{align} f(x) = \dfrac{\color{#3a9a38}{3}}{\color{#ec0000}{2x+2}} \color{#333fb1}{-2} \end{align}||
After factoring the denominator, the final result is: ||\begin{align} f(x) = \dfrac{\color{#3a9a38}{3}}{\color{#ec0000}{2(x+1)}} \color{#333fb1}{-2} \end{align}||