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) = {{x+1}\over{x-1}}# and #g# the function with rule #g(x)={{1}\over{x^2+2}}#.

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

#\left({f + g}\right)(x)\,=\, # #{{x^3+x^2+3\cdot x+1}\over{\left(x-1\right)\cdot \left(x^2+2\right) }}#

Indeed, #\left({f + g}\right)(x)=f(x)+g(x)=\left({{x+1}\over{x-1}}\right)+\left({{1}\over{x^2+2}}\right)={{x^3+x^2+3\cdot x+1}\over{\left(x-1\right)\cdot \left(x^2+2\right) }}#.

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