Respuesta :
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]
