Question

Find each of the following values given that ​f(x)equals StartFraction 1 Over StartRoot x plus 5 EndRoot EndFraction and ​g(x)equals 3x plus 1. a. ​(fplus​g)(negative 4​) b. ​(fminus​g)(0​) c. ​(ftimes​g)(negative 1​) d. (StartFraction f Over g EndFraction )​(1​)

Answer 1

a.
`(f+g)(-4)=-10`

b.
`(f-g)(0)=(1)/(√(5) ) -1`

c.
`(fg)(-1)=-1`

d.
`((f)/(g))(1)=(1)/(4√(6) )`

Explanation:

Given that:

`f(x)=(1)/(√(x+5) )`

`g(x)=3x+1`

The sum, rest, multiplication or division of functions are calculated as following:

`(f+g)(x)=f(x)+g(x)\\(f-g)(x)=f(x)-g(x)\\(fg)(x)=f(x)g(x)\\((f)/(g) )(x)=(f(x))/(g(x))`

For values of x where f(x) and g(x) are defined and in the case of
`((f)/(g) )(x)` for values of g(x) different from zero.

Taking into account that f(x) and g(x) are defined for values of x equals to -4, 0, 1 and -1 and g(1) is different from zero, we get:

`(f+g)(x)=f(x)+g(x)\\(f+g)(x)=(1)/(√(x+5) ) +3x+1\\(f+g)(-1)=(1)/(√(-4+5) ) +3(-4)+1\\(f+g)(-1)=-10`

`(f-g)(x)=f(x)-g(x)\\(f-g)(x)=(1)/(√(x+5) ) -(3x+1)\\(f-g)(0)=(1)/(√(0+5) ) -(3(0)+1)\\(f-g)(0)=(1)/(√(5) ) -1`

`(fg)(x)=f(x)g(x)\\(fg)(x)=(1)/(√(x+5) ) (3x+1)\\(fg)(-1)=((1)/(√(-1+5) ) )(3(-1)+1)\\(fg)(-1)=-1`

`((f)/(g))(x)=(f(x))/(g(x))  \\((f)/(g))(x)=(1)/((√(x+5))(3x+1)) \\((f)/(g))(1)=(1)/((√(1+5))(3(1)+1))\\((f)/(g))(1)=(1)/(4√(6) )`