Question

if f(x) = the square root of x-12 and g(x) = (1/x^2 -8) then what is the domain of ( g o f) (x) ?

Answer 1

\[f(x) = \sqrt{x - 12}\]\[g(x) = \frac{1}{x^2-8}\] g(f(x)) means put f(x) into g(x)\[\frac{1}{(\sqrt{x-12})^2 - 8}\] Squaring a square root cancels the square root leaving you with \[\frac{1}{x-12 - 8}\]Simplify \[g(f(x)) = \frac{1}{x - 20}\]

Answer 2

Now in order to find the domain of g(f(x)) you need to take into account the domain of f(x) and g(x)

Answer 3

Domain of f(x) is \[x \ge 12\] and the domain of g(x) is \[\left\{ x: x \ne \pm \sqrt{8} \right\}\]

Answer 4

how did you find those domains?

Answer 5

The domain of f(x) and g(x) ?

Answer 6

yes

Answer 7

Ok for the domain of f(x).. A square root function can contain any positive value (including 0) but not negative numbers (because those are complex numbers). So we set what is inside the square root to greater than or equal to zero like this \[x - 12 \ge 0\] and then solve for x \[x \ge 12\] This means f(x) can contain any value that is equal to 12 ore greater than 12.. If its below 12 then we get complex numbers and we dont want those.

Answer 8

For g(x) we cannot divide by 0 because its undefined. So we need to find what value makes the denominator 0. So i set the denominator equal to 0 like this \[x^2 - 8 = 0\] solve for x \[x^2 = 8\] \[\sqrt{x^2} = \pm \sqrt{8}\] \[x = \pm \sqrt{8}\] This means x cannot be positive sqrt(8) or negative sqrt(8) because these two numbers make the denominator zero.

Answer 9

you are super helpful!!!!

Answer 10

Now to find the domain of f(g(x))

Answer 11

Im looking up on how that is done so it may take me a little bit.

Answer 12

the domain of f(g(x)) is the set of all numbers x in the domain of g such that g(x) is in the domain of f

Answer 13

My guess would be \[[12, 20) \cup (20, \infty)\]

Answer 14

I was never good at these composite functions.

Answer 15

lol

Answer 16

haha thank you!!!!

Answer 17

is your homework online or do you have to write it on paper?

Answer 18

I dont think its right.. because we have g(f(x))

Answer 19

it's on paper. but i found the answer!!

Answer 20

What is it and maybe i can figure it out?

Answer 21

the answer is the intervals (15,25) and (30,50)

Answer 22

Wow, i have no idea how lol. I knew i hated these for a reason

Answer 23

it was multiple choice!!

Answer 24

so there can be many more answers

Answer 25

oh ok