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f(x)=3x2-3 g(x)=8/xI need help finding this answer f*g(x) and also g*f(x) I cant figure...

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tosassyas | Student, College Freshman | eNotes Newbie

Posted July 26, 2011 at 12:42 PM via web

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f(x)=3x2-3 g(x)=8/x

I need help finding this answer f*g(x) and also g*f(x) I cant figure this out

 

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4 Answers | Add Yours

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crounds | High School Teacher | eNoter

Posted July 28, 2011 at 6:29 AM (Answer #2)

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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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shaznl1 | High School Teacher | Salutatorian

Posted July 30, 2011 at 11:20 AM (Answer #3)

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f(g(x)= put an f(x) in where g(x) has an x in it

g(x)= 8/x

put in a f(x)

f(g(x)= 8/(3x^2-3)

g(f(x)) is the total opposite

f(x)= 3x^2-3

put in a g(x)

3(8/x)^2-3=3*64/x^2-3=192/x^2 -3

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dark0flame | Student , Undergraduate | eNoter

Posted August 25, 2011 at 11:12 PM (Answer #4)

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assuming you are talking about composite functions
if we substitute 'g(x)' with 'z' so that
g(x)=z

then need to solve for f(z)
so f(z)=3z^2-3

then substitute g(x) back in for z
we get f(g(x))=3(g(x))²-3

f(g(x))=3(8/x)²-3
=3(8²/x²)-3
=3(64/x²)-3
=192/x²-3

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
fg(x)
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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 26, 2011 at 3:57 PM (Answer #5)

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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.

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