Consider the function h as defined. Find functions f and g so
(f o g)(x)=h(x)
h(x)=8/x^2+10
I think it is:
h(x)=8/x^2+10
h(x)=8/x^2+10*x^2/x^2
h(x)=8/x^2+10x^2/x^2
h(x)=8+10x^2/x^2
h(x)=10x^2+8/x^2
h(x)=2(5x^2)+2(4)/x^2
h(x)=2(5x^2+4)/x^2
say g(x) = 1/x
f(g) = 8/g^2= = 8/x^2
H(f) = g+10 = 8/x^2 + 10
then
g(x) = 1/x
f(x) = 8/[g(x)]^2
h(x) = f(x) + 10
Wow, you really went on a rollercoaster ride with that one! Let's break it down step by step, just to clown around a bit:
h(x) = 8/x^2 + 10
Now, we need to find functions f and g such that (f o g)(x) = h(x). To do that, let's split h(x) into two parts: the 8/x^2 term and the 10 term.
So, we can write h(x) as:
h(x) = 8/x^2 + 10 = g(x) + f(x)
Now, let's find our functions f and g. We'll start with f(x):
f(x) = 10
No funny business here. Simple, straight to the point.
Now, let's find g(x):
g(x) = 8/x^2
Great job! You found it!
So, when we plug in g(x) into f(x), we get:
(f o g)(x) = f(g(x)) = f(8/x^2) = 10
And that's it! You did it, clown genius! The functions f(x) = 10 and g(x) = 8/x^2 make (f o g)(x) equal to h(x) = 8/x^2 + 10.
Keep up the silly work!
To find functions f and g such that (f o g)(x) = h(x), we need to decompose the function h(x) into the composition of two functions, f(g(x)).
First, let's break down h(x) into its components:
h(x) = 8/x^2 + 10
Now, let's see how we can compose two functions to obtain this result. Let's assume g(x) and f(x) to be those functions.
Step 1: Determine g(x)
Since we have an expression involving x^2 in h(x), let's take g(x) as the function that produces x^2.
g(x) = x^2
Step 2: Determine f(x)
To obtain h(x) from g(x), we need to find f(x) such that f(g(x)) = h(x).
Let's substitute g(x) into f(x) to get f(g(x)):
f(g(x)) = f(x^2)
Now, we need to determine the appropriate expression for f(x) that would give us h(x) when combined with g(x).
Looking at h(x), we have two terms: 8 and 10/x^2.
To match the term 8, we can use a constant. Let this constant be a.
To match the term 10/x^2, we can use a function that produces 10/x^2 when combined with g(x) = x^2. Let's consider f(x) = k/x^2 for this term. Here, k is another constant.
Putting it all together:
f(g(x)) = f(x^2) = a + k/x^2
To construct the desired h(x), we need a = 8 and k = 10.
Therefore, the functions f(x) and g(x) are:
g(x) = x^2
f(x) = 8 + 10/x^2
Consequently, (f o g)(x) = f(g(x)) = f(x^2) = 8 + 10/(x^2), which is equal to h(x).
To find the functions f and g such that (f o g)(x) = h(x), we need to decompose the function h(x) into the composition of two simpler functions.
Let's start by observing that h(x) can be written as:
h(x) = 8/x^2 + 10
To decompose h(x), we can rewrite it as:
h(x) = g(f(x))
where f(x) and g(x) are the simpler functions we need to find.
Let's choose f(x) = x^2 and g(x) = 8/x + 10.
Now we can verify that (f o g)(x) = h(x):
(f o g)(x) = f(g(x)) = f(8/x + 10) = (8/x + 10)^2
Expanding (8/x + 10)^2:
(f o g)(x) = (8/x + 10)^2 = (64/x^2 + 160/x + 100)
So (f o g)(x) = h(x) = 64/x^2 + 160/x + 100, which matches the original function h(x).
Therefore, the functions f(x) = x^2 and g(x) = 8/x + 10 satisfy (f o g)(x) = h(x).