Question

I Need to find the Function Operations​

Answer 1

Answer:

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

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

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

Step-by-step 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:

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