Question

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

Answer 1

The arithmetic combinations of given functions are (f + g)(x) = 2x, (f - g)(x) = 4, (f
`*` g)(x) =
`\mathrm{x}^(2)-4` ,
`\left((f)/(g)\right)(\mathrm{x})=(x+2)/(x-2)`

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
`*` g)(x),
`\left((f)/(g)\right)(\mathrm{x})`

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
`*` g)(x) = f(x)
`*` g(x)

= (x + 2)
`*` (x – 2)

= x
`*` (x – 2) + 2
`*` (x -2)

`=x^(2)-2 x+2 x-4`

`=x^(2)-4`

Therefore (f
`*` g)(x) =
`x^(2)-4`

now,

`\left((f)/(g)\right)(\mathrm{x})=(f(x))/(g(x))`

`=(x+2)/(x-2)`

`((f)/(g))(x)` =
`(x+2)/(x-2)`

Hence arithmetic combinations of given functions are (f + g)(x) = 2x, (f - g)(x) = 4, (f
`*` g)(x) =
`\mathrm{x}^(2)-4` ,
`\left((f)/(g)\right)(\mathrm{x})=(x+2)/(x-2)`