F(x) = √(x+7) -√(x^2+2x-15) find the domain

Question

F(x) =
`√(x+7) -√(x^2+2x-15)` find the domain

Answer 1

f(x) = 
find the domain

Answer 2

x >= -7 ................(1a)

x >= 3 ...............(2a1)

Explanation:

f(x) =
`√(x+7)-√(x^2+2x-15)` .............(0)

find the domain.

To find the (real) domain, we need to ensure that each term remains a real number.

which means the following conditions must be met

x+7 >= 0 .....................(1)

AND

x^2+2x-15 >= 0 ..........(2)

To satisfy (1), x >= -7 .....................(1a) by transposition of (1)

To satisfy (2), we need first to find the roots of (2)

factor (2)

(x+5)(x-3) >= 0

This implis

(x+5) >= 0 AND (x-3) >= 0.....................(2a)

OR

(x+5) <= 0 AND (x-3) <= 0 ...................(2b)

(2a) is satisfied with x >= 3 ...............(2a1)

(2b) is satisfied with x <= -5 ................(2b1)

Combine the conditions (1a), (2a1), and (2b1),

x >= -7 ................(1a)

AND

(

x >= 3 ...............(2a1)

OR

x <= -5 ................(2b1)

)

which expands to

(1a) and (2a1) OR (1a) and (2b1)

( x >= -7 and x >= 3 ) OR ( x >= -7 and x <= -5 )

Simplifying, we have

x >= 3 OR ( -7 <= x <= -5 )

Referring to attached figure, the domain is indicated in dark (purple), the red-brown and white regions satisfiy only one of the two conditions.