Question

Precalculus help!

1. Find (fog)(x) when f(x)=2/x+3 and g(x)=1/2x.

2. Find (f*g)(x) when f(x)=(sqrt)x+3/x and g(x)=(sqrt)x+3/2x.

3. Find (f+g)(x) when f(x)=x^3-2x^2+1 and g(x)=4x^3-5x+7

Answer 1

1)
`(f o g)(x)=(4)/(x+6)`

2)
`(f*g)(x)=\frac{2x^2+9x^{(3)/(2)}+9}{2x^2}`

3)
`(f+g)(x)=5x^3-2x^2-5x+8`

Explanation:

1) Given :
`f(x)=(2)/(x+3)` and
`g(x)=(1)/(2)x`

To find :
`(f o g)(x)`

Solution :

We know that,

`(f o g)(x)=f(g(x))` i.e g(x) within f(x),

`f(g(x))=(2)/(g(x)+3)`

`f(g(x))=(2)/((1)/(2)x+3)`

`f(g(x))=(2)/((x+6)/(2))`

`f(g(x))=(4)/(x+6)`

Therefore,
`(f o g)(x)=(4)/(x+6)`

2) Given :
`f(x)=\sqrt x+(3)/(x)` and
`g(x)=\sqrt x+(3)/(2x)`

To find :
`(f*g)(x)`

Solution :

We know that,

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

`f(x) * g(x)=(\sqrt x+(3)/(x))* (\sqrt x+(3)/(2x))`

`f(x) * g(x)=x+(3\sqrt x)/(2x)+(3\sqrt x)/(x)+(9)/(2x^2)`

`f(x) * g(x)=(2x^2+3x\sqrt x+6x\sqrt x+9)/(2x^2)`

`f(x) * g(x)=\frac{2x^2+9x^{(3)/(2)}+9}{2x^2}`

Therefore,
`(f*g)(x)=\frac{2x^2+9x^{(3)/(2)}+9}{2x^2}`

3) Given :
`f(x)=x^3-2x^2+1` and
`g(x)=4x^3-5x+7`

To find :
`(f+g)(x)`

Solution :

We know that,

`(f+g)(x)=f(x)+g(x)`

`f(x)+g(x)=x^3-2x^2+1+4x^3-5x+7`

`f(x)+g(x)=5x^3-2x^2-5x+8`

Therefore,
`(f+g)(x)=5x^3-2x^2-5x+8`