Question

I Need to find the Function Operations​

Answer 1

1)
`(f+g)(x)=2x^2+3x-6`

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

3)
`(f \cdot g)(x)=2x^3-4x^2-9x+9`

Explanation:

So we have the two functions:

`f(x)=2x^2+2x-3\text{ and } g(x)=x-3`

And we want to find (f+g)(x), (f-g)(x), and (f*g)(x).

1)

(f+g)(x) is the same to f(x)+g(x). Substitute:

`(f+g)(x)=f(x)+g(x)\\=(2x^2+2x-3)+(x-3)`

Combine like terms:

`=(2x^2)+(2x+x)+(-3-3)`

Add:

`=2x^2+3x-6`

So:

`(f+g)(x)=2x^2+3x-6`

2)

(f-g)(x) is the same to f(x)-g(x). So:

`(f-g)(x)=f(x)-g(x)\\=(2x^2+2x-3)-(x-3)`

Distribute:

`=(2x^2+2x-3)+(-x+3)`

Combine like terms:

`=(2x^2)+(2x-x)+(-3+3)`

Simplify:

`=2x^2+x`

So:

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

3)

(f*g)(x) is the same to f(x)*g(x). Thus:

`(f\cdot g)(x)=f(x)\cdot g(x)\\=(2x^2+2x-3)(x-3)`

Distribute:

`=(2x^2+2x-3)(x)+(2x^2+2x-3)(-3)`

Distribute:

`=(2x^3+2x^2-3x)+(-6x^2-6x+9)`

Combine like terms:

`=(2x^3)+(2x^2-6x^2)+(-3x-6x)+(9)`

Simplify:

`=2x^3-4x^2-9x+9`

So:

`(f \cdot g)(x)=2x^3-4x^2-9x+9`