The Sum of Functions

Operations on functions are performed the same way operations on numbers are performed.

The domain of the sum of functions corresponds to the intersection of the domains of the functions in question. If there is a denominator, the restrictions on it must be included.

Function |k| is defined by |k(x)=x+1| and function |l| is defined by |l(x)=2x+1.| The sum of the functions results in the following.
||\begin{eqnarray*} (k+l)(x)&=&k(x)+l(x) \\
           &=& (x+1)+(2x+1) \\
           &=& 3x+2 \end{eqnarray*}||

Function |k|’s domain corresponds to |\mathbb{R}.| Function |l|’s domain also corresponds to |\mathbb{R}.| Therefore, the function given by |k+l|’s domain will correspond to the intersection of the two initial domains. Function |k+l|’s domain is |\mathbb{R}.|

Function |i| is defined by |i(x)=x+2| and function |j| is defined by |j(x)=\sqrt{x}.| The sum of the functions results in the following.
||\begin{eqnarray*} (i+j)(x) &=& i(x)+j(x) \\
&=&(x+2)+\sqrt{x} \\
&=& x+\sqrt{x}+2 \end{eqnarray*}||

Function |i|’s domain corresponds to |\mathbb{R}| and function |j|’s domain corresponds to |\mathbb{R}^{+}.| The function’s domain |i+j| will then correspond to the intersection of the two initial domains. Therefore, function |i+j|’s domain is |\mathbb{R}^{+}.|

Function |f| is defined by |f(x)=\dfrac{2}{x}| and function |g| is defined by |g(x)=2x.| The sum of the functions results in the following. ||\begin{align} (f+g)(x) &= f(x) + g(x) \\ &=\dfrac{2}{x} + 2x \\ &= \dfrac{2}{x} +\dfrac{2x^2}{x} \\ &= \dfrac{2+2x^2}{x} \\ &= \dfrac{2(1+x^2)}{x} \end{align}||

Function |f|’s domain is |\mathbb{R} \backslash \lbrace 0 \rbrace| and function’s domain |g|’s domain is |\mathbb{R}.| Therefore, the domain of their sum is |\mathbb{R} \backslash \lbrace 0 \rbrace \cap \mathbb{R} = \mathbb{R} \backslash \lbrace 0 \rbrace.|

Graphically, the sum of functions is obtained by adding the ranges of the functions of the problem.

In the first example, if we make a table of values for the functions |k(x)=x+1,| |l(x)=2x+1,| and their sum |k+l,| the result is the following.

Function |k+l| is increasing and its domain is |\mathbb{R}.|

To obtain the graph of a sum of functions, add the ranges of each of the functions.

The function |i+j| is an increasing function whose domain is |\mathbb{R}^{+}.|

Function |f| is defined by |f(x)= {\mid}x{\mid}| and function |g| is defined by |g(x)=x^2.|