## Content

### The arithmetic of functions

Just as numbers can be added, subtracted, multiplied and divided to form other numbers, so functions can be added, subtracted, multiplied and divided to form other functions.

We have already seen in the TIMES module Polynomials (Year 10) that polynomials can be added, subtracted, multiplied and divided. The definitions given there are consistent with the definitions for functions in general.

In later sections of this module, we will look at two other ways of forming new functions: composition of functions and finding the inverse of a function.

Throughout this module, we consider **real functions**, that is, functions \(f \colon D \to \mathbb{R}\), where the domain \(D\) is a subset of \(\mathbb{R}\).

#### Addition and subtraction of functions

If \(f\) and \(g\) are functions with domains \(D_f\) and \(D_g\), then the **sum of the functions** \(f + g\) is defined by

The domain of \(f+g\) is \(D_f \cap D_g\). That is, the function \(f+g\) is defined only where both \(f\) and \(g\) are defined.

For example, let \(f(x) = e^x\) and \(g(x) = \dfrac{1}{x}\), where the domain of \(f\) is \(\mathbb{R}\) and the domain of \(g\) is \(\mathbb{R} \setminus \{0\}\). Then the function \(f + g\) has domain \(\mathbb{R} \cap (\mathbb{R} \setminus \{0\}) = \mathbb{R} \setminus \{0\}\), with

\[ (f + g)(x) = e^x + \dfrac{1}{x}. \]Similarly, if \(f\) and \(g\) are functions with domains \(D_f\) and \(D_g\), then the **difference of the functions** \(f - g\) is defined by

The domain of \(f - g\) is \(D_f \cap D_g\).

#### Products and quotients of functions

If \(f\) and \(g\) are functions with domains \(D_f\) and \(D_g\), then the **product of the functions** \(f \cdot g\) is defined by

The domain of \(f \cdot g\) is \(D_f \cap D_g\).

For example, using the functions \(f(x) = e^x\) and \(g(x) = \dfrac{1}{x}\) from above, we have

\[ (f \cdot g)(x) = e^x \cdot \dfrac{1}{x} = \dfrac{e^x}{x}. \]If \(f\) and \(g\) are functions with domains \(D_f\) and \(D_g\), then the **quotient of the functions** \(\dfrac{f}{g}\) is defined by

The domain of \(\dfrac{f}{g}\) is \(\{\, x \in D_f \cap D_g : g(x) \neq 0 \,\}\).