Arithmetic operations for functions

Functions: Introduction to functions

Arithmetic operations for functions

We discuss five methods of creating a new real function out of two existing real functions.

Arithmetic operations on functions

Let $f$ and $g$ be real valued functions with the same domain and let $a$ be a real number. We define

the sum function $f+g$ as the function with rule $(f+g)(x) = f(x)+g(x)$ ;
the scaled function $a\cdot f$ as the function with rule $(a\cdot f)(x) = a\cdot{f(x)}$ ;
the difference function $f-g$ as the function $f+(-1)\cdot g$ ;
the product function $f\cdot g$ as the function with rule $(f\cdot g)(x) = f(x)\cdot g(x)$ ;
the quotient function $\frac{f}{g}$ as the function with rule $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$ .

Because all five are determined by arithmetic operations on values of the corresponding function, these operations are called arithmetic.

If $a$ is a number and $f$ is a function, the function $a\cdot f$ has value $a\cdot f(x)$ at $x$ . If we consider $a$ as a constant function (which assigns the value $a$ to every $x$ , so $a(x) = a$ ), then $(a\cdot f) (x) = a \cdot f(x)$ . So there is no confusion possible between the two interpretations of $a$ .

A special instance of the quotient function is $\dfrac{1}{f}$ , the function defined by $\dfrac{1}{f} (x)= \dfrac{1}{f(x)} = f(x)^{-1}$ . It is risky to write $f^{-1}$ for this function, since this notation is commonly used for the inverse function of $f$ , which will be defined later.

Three kinds of special functions

Let $a$ be a real number. The function $C_a$ with rule $C_a(x)=a$ is called the constant function $a$ . If $f$ is a function, then $C_a\cdot f = a \cdot f$ . After all, $(C_a\cdot f)(x)= C_a(x)\cdot f(x) = a \cdot f(x) = (a\cdot f)(x)$ for each $x$ from the domain of $f$ .
The absolute value is a real function with rule $| x | = \begin{cases} \phantom{-}x & \mbox{ if }\ x \ge 0\\ -x & \mbox{ if }\ x<0\end{cases}$ This function can also be written as $\sqrt{x^2}$ .
A function $f$ with a rule of the form $f(x)=a\cdot x+b$ for two numbers $a$ and $b$ with $a\ne0$ , is called a linear function. A special case presents itself when $a=1$ and $b=0$ : then $f(x) = x$ applies; this is the identity.

Frequently the constant function $a$ instead of $C_a$ is mentioned. Beware: $a(x+1)$ stands for $ax+a$ and not for the value at $x+1$ of the constant function $C_a$ , which would be $a$ . Hence $C_a(x+1)$ is not the same as $a(x+1)$ . In order to avoid confusion, we prefer to make the product symbol $\cdot$ visible and write $a\cdot(x+1)$ instead of $a(x+1)$ .

If we would take $a=0$ in the rule $ax+b$ for a linear function, then we get the constant function $b$ . This is the reason for excluding $a=0$ in the definition of a linear function.

Let $f$ be the function with rule $f(x) = 1$ and $g$ the function with rule $g(x)={{x+1}\over{x-1}}$ .

Determine the function rule for ${f + g}$ .

$\left({f + g}\right)(x)\,=\,$ ${{2\cdot x}\over{x-1}}$

Indeed, $\left({f + g}\right)(x)=f(x)+g(x)=\left(1\right)+\left({{x+1}\over{x-1}}\right)={{2\cdot x}\over{x-1}}$ .

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