Find the arithmetic combination of f(x) =x+2, g(x)=x-2

Answer:The arithmetic combinations of given functions are (f + g)(x) = 2x, (f - g)(x) = 4,  (f [tex]\times[/tex] g)(x) = [tex]\mathrm{x}^{2}-4[/tex] , [tex]\left(\frac{f}{g}\right)(\mathrm{x})=\frac{x+2}{x-2}[/tex]Solution:   Given, two functions are f(x) = x + 2 and g(x) = x – 2 We need to find the arithmetic combinations of given two functions .Arithmetic functions of f(x) and g(x) are (f + g)(x), (f – g)(x), (f [tex]\times[/tex] g)(x), [tex]\left(\frac{f}{g}\right)(\mathrm{x})[/tex]Now, (f + g)(x) = f(x) + g(x) = x + 2 +x – 2 = 2x Therefore (f + g)(x) = 2x similarly,(f - g)(x) = f(x) - g(x) = x + 2 –(x – 2) = x + 2 –x + 2 = 4 Therefore (f - g)(x) = 4 similarly,(f [tex]\times[/tex] g)(x) = f(x) [tex]\times[/tex] g(x)= (x + 2) [tex]\times[/tex] (x – 2)= x [tex]\times[/tex] (x – 2) + 2 [tex]\times[/tex] (x -2)[tex]=x^{2}-2 x+2 x-4[/tex][tex]=x^{2}-4[/tex]Therefore  (f [tex]\times[/tex] g)(x) = [tex]x^{2}-4[/tex]now,[tex]\left(\frac{f}{g}\right)(\mathrm{x})=\frac{f(x)}{g(x)}[/tex][tex]=\frac{x+2}{x-2}[/tex][tex](\frac{f}{g})(x)[/tex] = [tex]\frac{x+2}{x-2}[/tex]Hence arithmetic combinations of given functions are (f + g)(x) = 2x, (f - g)(x) = 4,  (f [tex]\times[/tex] g)(x) = [tex]\mathrm{x}^{2}-4[/tex] , [tex]\left(\frac{f}{g}\right)(\mathrm{x})=\frac{x+2}{x-2}[/tex]