Consider the function h as defined.Find function f and g so that (f o g)(x) = h(x). h(x) = 1/x^2 - 3?

Question

College algebra

Answer 1

f(x): 1/x-3
g(x): x^2

f(g(x)) means taking g(x) and plugging it into f(x). So, we need to find an f(x) and g(x) that produce the fog(x) when you plug g(x) into f(x). Let's pick an easy one for f:

f(x) = 1/x-3

But h(x) = f(g(x)), so
f(g(x))=1/g(x)-3

What must g(x) be? From what we know of h(x), it has to be x^2


Hope this helps.

Answer 2

Quad and ksoileau have already exceptionally lots overwhelmed the main out of this question. So enable me basically make a pair of feedback: a) in case you require continuity, then there is strictly one function that fulfill your difficulty. it is because of the fact i) Your difficulty implies immediately that f is invertible (f^{-a million}(x) = f(f(x))) so ii) f is a homeomorphism of the particular line and f preserves order (because of the fact the actual line is an ordered set.) so iii) if f(a) = b for b no longer equivalent to a for some a, then we can, without loss of generality, anticipate that f(a) < a. Then f(f(a)) < f(a) < a, and further f(f(f(a))) < a. b) As quad has shown, it suffices to have one discontinuity. c) we can take additionally ksoileau's physique of innovations slightly extra. Given a function f, we can evaluate the curve C in R^3 given by (x,f(x),f(f(x))). Requiring that f(f(f(x))) = x is then the comparable as requiring that C is invariant below the action of the rotation matrix [ 0 0 a million ] [ a million 0 0 ] [ 0 a million 0 ] [ 0 0 a million ] [ a million 0 0 ] [ 0 a million 0 ] [ 0 0 a million ] [ a million 0 0 ] [ 0 a million 0 ] and requiring that f to be a function ability that the intersection of C with any of the planes given by {x = consistent} {y = consistent} {z = consistent} has a single element. that's, in some experience, the graphical illustration of the scheme. f is now represented by the rotation matrix. We observe that we've limited freedom in pick the curve C. A sufficiently widespread shape (that maximum easily does no longer get well all available purposes) is here: the matrix above generates a discrete subgroup G (with 3 components) of O(3), that fixes the subspace spanned by the vector (a million,a million,a million). subsequently we can build the orbifold O(3)/G, which we call the elementary area. on the orbifold we can basically approximately freely prescribe a Curve: we basically could make certain that the curve C defines a function. a standard thank you to make certain that C defines a function is to %. a illustration of the orbifold (or a subset therefor) this is "sturdy". as an occasion, we can limit to the slab { -infinity < x < 0, a (a,infty) g: (-infty,0) -> (0,a) Then the function f given by f (x) = h(x) if x < 0 g(h^{-a million}(x)) if x > a g^{-a million}(x) if 0

Answer 3

Not so tough..

Just think of the order that you do things to evaluate h(x). First you square x. so make g(x) = x^2, then you reciprocate and subtract 3

so f(y) = 1/y-3

in the above let y=g(x), do the substitution.

Note that it's not unique.

You could take g(x) = 1/x^2, and f(y) = y-3.

Answer 4

If fog(x) = h(x) = 1/x^2 - 3, the you could
Let f(x) = 1/x^2-3, yes it is the same as h(x)
And let g(x)=x
Then f(g(x)) = 1/[g(x)]^2 - 3 = 1/x^2 - 3 = h(x)