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I'll explain to you how to solve this problem both cases: multiplication or composition of functions.
Multiplication of functions:
Since f(x)*g(x) = g(x)*f(x), you need to calculate the product only once.
f(x)*g(x) = (3x^2 - 3)*(8/x)
Open the brackets => f(x)*g(x) =24x^2/x - 24/x
f(x)*g(x) = 24x - 24/x
Composition of functions.
Since f(x)*g(x) is different from g(x)*f(x), you need to determine both results.
f(x)*g(x) = f(g(x)) = f(8/x)
Do the intuitive switch by replacing x by 8/x in the expression of f(x) .
f(8/x) = 3*(8/x)^2 - 3 => f(8/x) = 192/x^2 - 3
Compose g(x) and f(x): g(f(x)) = g(3x^2 - 3)
g(3x^2 - 3) = 8/(3x^2 - 3)
Notice the difference between the results of compositions of functions f and g. All depends on the order of entrance of the functions in the composition operation.
assuming you are talking about composite functions
if we substitute 'g(x)' with 'z' so that
then need to solve for f(z)
then substitute g(x) back in for z
we get f(g(x))=3(g(x))²-3
it is useful to note that the whole f(g(x)) got a little tiresome for mathematicians... so they decided to drop the parenthesis between the functions.. giving
we can solve composite functions with many simple functions like this...
and once you understand the concept, you can 'decluster' all the functions in one step... but understanding how it works first is critical
f(g(x)= put an f(x) in where g(x) has an x in it
put in a f(x)
g(f(x)) is the total opposite
put in a g(x)
If "*" is indicating multiply you need to mulitiply the two functions together:
(3x^2 - 3)(8/x) original problem
3x^2(8/x) - 3(8/x) distirbutive property
(3/8)x - 24/x multiply
I have a feeling though that your problem is supposed to be a compostion, like a cake batter. As you put in more ingredients (equation) you get a new batter (new equation). Lets try this with a number first, say x = 4.
f(g(x) = f(g(4)) using the order of operations, we must find the value of g(4) first (every place in the g equation put the number 4 and simplify) g(4) = 8/4 = 2
now that value goes into the f equation: f(2) = 3(2)^2-3 = 12 - 3 = 9
so f(g(3)) = 9
when you use equations/expressions you do the same thing
f(g(x)) (1) g(x) = 8/x (plug this expression into the f equation everywhere you have an x and simplify)
(2) f(8/x) = 3(8/x)^2 - 3 = 192/x^2 - 3
g(f(x)) (1) f(x) = 3x^2 - 3 (plug this expression into the g equation everywhere you have an x and simplify)
(2) g(3x^2 - 3) = (8(3x^2 - 3))/x
= (24x^2 -24) / x
notice these answers are NOT the same, if they were it means the two original functions are inverses of each other. What you have is a line and a reciprocal function (hyperbola)
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