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

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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.

dark0flame | Student, Undergraduate | (Level 3) eNoter

Posted on

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

shaznl1 | High School Teacher | (Level 1) Salutatorian

Posted on

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

crounds | High School Teacher | (Level 1) eNoter

Posted on

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)