Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column.
If [latex]w\left(y\right)[/latex] is the wife’s income and [latex]h\left(y\right)[/latex] is the husband’s income in year [latex]y[/latex], and we want [latex]T[/latex] to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
[latex]T=h+w[/latex]
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and numerical outputs.
algebraic operations on functions
For two functions [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)[/latex] with real number outputs, we define new functions [latex]f+g,f-g,f\cdot{g}[/latex], and [latex]\dfrac{f}{g}[/latex] by the relations:
The domain [latex](g-f)(x)[/latex] is all real numbers and the domain of [latex]\left(\dfrac{g}{f}\right)(x)[/latex] is [latex]x\ne1[/latex]. The functions are not the same.
Note: For [latex]\left(\dfrac{g}{f}\right)\left(x\right)[/latex], the condition [latex]x\ne 1[/latex] is necessary because when [latex]x=1[/latex], [latex]f(x)=1-1=0[/latex], which makes the quotient function undefined (dividing by zero).