Composite Functions

[mathxyz]

Find f + g, f - g, fg and f/g and state their domains.

1) f(x) = sqrt{1 + x}, g(x) = sqrt{1 - x}

MY WORK FOR f + g:

Is this correct:

f + g = 1 + x + 1 - x

f + g = 2? What is the domain? I have no clue.

How do I solve the rest? How do I find their domains?

 

[soroban]

Hello, mathxyz!

I assume you know what a domain is: .the set of all "legal" values of x.

Find f + g, f - g, fg and f/g and state their domains.

1) f(x) = sqrt{1 + x}, g(x) = sqrt{1 - x}

(a) . f + g .= .sqrt{1 + x} + sqrt{1 - x}

We cannot have the square root of a negative quantity.
. . So: . 1 + x <u>></u> 0 . ---> . x <u>></u> -1
. . and . 1 - x <u>></u> 0 . ---> . x <u><</u> 1

Hence, x is <u>between</u> -1 and +1 (including the endpoints).

. . . Domain: . -1 <u><</u> x <u><</u> 1


(b) .f - g .= .sqrt{1 + x} - sqrt{1 - x}

. . This has the same domain as (a).


(c) .fg .= .sqrt{1 + x}·sqrt{1 - x} .= .sqrt{1 - x²}

. . 1 - x² <u>></u> 0 . ---> . x² <u><</u> 1 . ---> . |x| <u><</u> 1 . ---> . -1 <u><</u> x <u><</u> 1


. . . . .f . . . . .sqrt{1 + x}
(d) .--- . = . ---------------
. . . . g . . . . .sqrt{1 - x}

This function seems to have the same domain as (a), (b) and (c) . . .
. . but if x = 1, we have a zero in the denominator . . . a no-no.

Therefore, the domain is: . -1 <u><</u> x < 1

 

[mathxyz]

Soroban

Soroban,

Thank you for your help. Did you get my questions about tangent and secant of the line?