Does anyone know how to solve this?

Question

Compute (f composite g)(x) and (g composite f)(x) when f(x)= square root of x and g(x)=x^4 -2x^2 +1

Answer 1

alright, so in order to do these composite things, you have to know the notation for functions.

when you have f(x) = ......., then that means put an 'x' in the function. This is the standard notation. If you had f(2), then that means put a '2' in the function in place of the 'x'.
So if you had f(2x), then that means instead of 'x', you would put '2x' in the function.
Now, instead of putting f(2x), you can create a new function that is called g(x) = 2x. and then f(2x) = f(g(x)). which means you would put g(x), whatever the function is, inside f(x) in place of x.

So in this case, g(x)=x^4 - 2x^2 + 1, so its basically asking you for f(x^4 - 2x^2 + 1) so you would have to put 'x^4 - 2x^2 + 1' inside the f(x) function in place of the x. f(x) = sqrt(x), so then (f composite g)(x), which is the same thing as f(g(x)) is equal to

(f comp g)(x) = sqrt (x^4 - 2x^2 + 1)
You can simplify this because if you let y = x^2, then this equation is (y^2 - 2y + 1), which is (y-1)^2 and sqrt of that is y-1, which is x^2-1. and this can also be factored into (x+1)(x-1)

so (f comp g)(x) = x^2-1 = (x+1)(x-1)


Similarly, if you wanted (g comp f)(x), then you would put the f function in the g function in place of the x. you would put 'sqrt x' inside the g function in place of x.

g(f(x)) = (sqrt x)^4 - 2(sqrt x)^2 + 1

now, (sqrt x)^2 is just x, and (sqrt x)^4 is x^2, so you get
x^2 - 2x + 1, which can be factored into (x-1)^2.

so (g comp f)(x) = (x-1)^2
and (f comp g)(x) = (x+1)(x-1)

Hope that helped you.

Answer 2

(f composite g)(x) = square root of g(x)= | x^2-1|
(g composite f)(x) = x^2 -2x+1

Answer 3

f(g(x))= the square root of (x^4 - 2x^2+1)
g(f(x))= x^2-2x+1

Answer 4

f(g(x)) = sqr root of x^4-2x^2+1
= sqr root of (x^2-1)^2
= x^2 - 1

g(f(x)) = (sqr root of x)^4 - 2 (sqr root of x)^2 +1
= x^2 - 2x + 1
= (x - 1) ^2

Answer 5

f(g(x))=sqrt(x^4-2x^2+1)]
g(f(x))= x^2-2x+1

Answer 6

g(x) = (x^2 - 1)^2
f(x) = sqrt(x)

fog(x) = sqrt(g) = x^2 - 1
gof(x) = (f^2 - 1)^2 = (x-1)^2

Answer 7

My brain! My brain!


ooooh... the agony....