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Dividing functions

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Dividing Functions

Dividing functions forms a new function from the quotient of two functions: (f/g)(x) equals f(x) divided by g(x). Learn the quotient rule, why the divisor cannot be zero and how that restricts the domain, and a worked example that factors and simplifies.

What dividing functions means

Dividing functions means forming a new function from the quotient of two existing functions. If you have two functions f and g, their quotient is written (f/g)(x) and is defined as f(x) divided by g(x). It is one of the four basic operations with functions, alongside adding, subtracting, and multiplying functions.

Dividing functions: the quotient rule The quotient of two functions f and g is written (f/g)(x) = f(x) / g(x), defined only where g(x) is not zero. Example: with f(x) = x squared minus 1 and g(x) = x plus 1, the quotient simplifies to x minus 1, valid for x not equal to negative 1. ( f / g )(x) = f(x) ÷ g(x) restriction: g(x) ≠ 0 worked example f(x) = x² − 1, g(x) = x + 1 (f/g)(x) = (x² − 1)/(x + 1) = x − 1, x ≠ −1
Dividing functions: (f/g)(x) = f(x) ÷ g(x), defined where g(x) is not zero.

The rule is simply (f/g)(x) = f(x)/g(x). To evaluate it at a number, divide the two output values: if f(3) = 12 and g(3) = 4, then (f/g)(3) = 12/4 = 3. To find the quotient as a formula, write f(x) over g(x) and simplify.

Dividing brings one important rule: you can never divide by zero. So the quotient (f/g)(x) is undefined wherever g(x) = 0, and every such x must be removed from the domain — even if the simplified formula seems to allow it. Checking which inputs are allowed connects directly to the domain and range of a function.

Let f(x) = x² − 1 and g(x) = x + 1. Then (f/g)(x) = (x² − 1)/(x + 1). The numerator factors as (x + 1)(x − 1), so the expression simplifies to x − 1. But because g(−1) = 0, the value x = −1 is still excluded: the final answer is x − 1 for all x ≠ −1.

  1. Write the quotient as f(x) over g(x).
  2. Simplify the expression if possible (factor and cancel).
  3. State the domain: exclude every x where the original g(x) = 0.

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