Combinations and Compositions of Functions: Learn It 1

  • Use algebraic operations to combine functions and create new expressions
  • Build a new function by combining two or more functions together
  • Calculate the output for composite functions for given values and determine the set of inputs that work for these functions
  • Break down a composite function into the original functions that were combined to make it

Combining Functions Using Algebraic Operations

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.

[latex]T\left(y\right)=h\left(y\right)+w\left(y\right)[/latex]

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 also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.

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:

 

[latex]\begin{align} \text{Addition:} \quad \left(f+g\right)\left(x\right) &= f\left(x\right)+g\left(x\right) \\[2mm] \text{Subtraction:} \quad \left(f-g\right)\left(x\right) &= f\left(x\right)-g\left(x\right) \\[2mm] \text{Multiplication:} \quad \left(f\cdot{g}\right)\left(x\right) &= f\left(x\right)\cdot{g}\left(x\right) \\[2mm] \text{Division:} \quad \left(\dfrac{f}{g}\right)\left(x\right) &= \dfrac{f\left(x\right)}{g\left(x\right)}, \quad \text{where } g(x) \neq 0 \end{align}[/latex]
Find and simplify the functions

[latex](g-f)(x)[/latex] and [latex]\left(\dfrac{g}{f}\right)(x)[/latex],

given

[latex]f\left(x\right)=x - 1[/latex] and [latex]g\left(x\right)={x}^{2}-1[/latex].

Give the domain of your result. Are they the same function?