Given f(x)=x2^2+2x and g(x)=1−x^2 , find f+g , f−g , fg , and fg . (f+g)(x)=(f-g)(x)=fg(x)=f/g(x)=**a ^ symbol indicates an exponent. example f(x)=x^2. 2 would be the exponent

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ZyairahW758274 ZyairahW758274 · Answer 1 · 2022-10-28T10:25:59+02:00

Explanation

In this problem, we have the following functions;

[tex]\begin{gathered} f(x)=x^2+2x, \\ g(x)=1-x^2. \end{gathered}[/tex]

We must find the functions:

0. (f + g)(x)

,

1. (f - g)(x)

,

2. (f g)(x)

,

3. (f/g)(x)

(1) (f + g)(x)

This function is given by:

[tex](f+g)(x)=f(x)+g(x)=(x^2+2x)+(1-x^2)=2x+1.[/tex]

(2) (f - g)(x)

This function is given by:

[tex](f-g)(x)=f(x)-g(x)=(x^2+2x)-(1-x^2)=2x^2+2x-1.[/tex]

(3) (f g)(x)

This function is given by:

[tex](f\cdot g)(x)=f(x)\cdot g(x)=(x^2+2x)\cdot(1-x^2)=x^2-x^4+2x-2x^2=-x^4-x^2+2x.[/tex]

(4) (f/g)(x)

This function is given by:

[tex](f/g)(x)=\frac{f(x)}{g(x)}=\frac{x^2+2x}{1-x^2}.[/tex]Answer

(1)

[tex](f+g)(x)=f(x)+g(x)=2x+1[/tex]

(2)

[tex](f-g)(x)=f(x)-g(x)=2x^2+2x-1[/tex]

(3)

[tex](f\cdot g)(x)=f(x)\cdot g(x)=-x^4-x^2+2x[/tex]

(4)

[tex](f/g)(x)=\frac{f(x)}{g(x)}=\frac{x^2+2x}{1-x^2}[/tex]