Question

Help! I'll fan and medal!
Find f(x) and g(x) so the function can be expressed as y = f(g(x)).

y = Two divided by x squared. + 3

Answer 1

what school do you go to?

Answer 2

Homeschool

Answer 3

do you use K12 virtual acadamey materials?

Answer 4

Ok are there answer choices, i want to make sure this answer is in range...

Answer 5

??? And no it's an essay question

Answer 6

OH its an essay question?

Answer 7

Yes unfortunately

Answer 8

\[y= \frac{ 2 }{ x ^{2} } + 3\]

Answer 9

That is the answer, can you give any instructions for the essay or do you even have them?

Answer 10

That's all it gave

Answer 11

ok...
I really don't know abou the essay part, I can't help you there sorry. :(

Answer 12

f( g(x) ) means replace any "x" in your equation with g(x)

for example *IF* you had at the rule:
\[ f(x) = \frac{2}{x} +3 \]
then
f( g(x) ) would be:
\[ f( g(x))= \frac{2}{g(x)} +3 \]

what should you make g(x) be ? i.e. what "rule"
so that if you replace g(x) with its "rule" you get
\[ f(g(x)) = \frac{2}{x^2} +3 \]

Answer 13

Is that what the answer is? @phi

Answer 14

@phi

Answer 15

@calculusxy

Answer 16

it's almost the answer

Answer 17

I'm confused at what to do, that didn't make sense to me @phi

Answer 18

The first thing is to get this idea:

"x" is a "placeholder", and it is something that shows where you "put something"

Think of the equation (for example)
f(x)= x+3
as a "pattern" or string of letters and numbers. But everywhere there is the symbol "x", you can replace it with a "new" letter or number or string.

normally, we use numbers, and we use the syntax
f(3)
to mean: replace x with 3 in f(x)=x+3
we would get: f(3) = 3+3

Answer 19

another example

g(x)= x+x+x

what is g(1) ?
we start with g(x)= x+x+x
we replace x with 1: g(1) = 1+1+1

Answer 20

once that idea makes sense, and you can do problems like

f(x)= x^2 +2x+1
f(2) = 2^2 +2*2+1
(which we can then simplify using "order of operations" and arithmetic)

we can tackle your problem

f( g(x) )

Answer 21

Let's say
f(x) = x+1

what is f( g(x) ) ?
we do the same thing as before: we replace the symbol x with g(x) :
f(x) = x+1 becomes
f( g(x) ) = g(x) + 1

in other words, everywhere you see "x" you erased it, and put in "g(x)"
the process is simple.

Of course, we don't get a number: g(x)+1 looks a bit strange, but that is ok.

Answer 22

but what if we had the equation
g(x) = x^2 + 2x + 1
??

g(x) is the short-name for the expression x^2 + 2x + 1

if we were told
f(x) = g(x)
we would know
f(x) = x^2 + 2x + 1

or if we knew
f(x) = g(x) +1
then we know
f(x) = x^2 + 2x + 1 + 1

Answer 23

now your problem:

Find f(x) and g(x) so the function can be expressed as y = f(g(x)).
\[ y= \frac{2}{x^2} +3 \]

we have to find f(x) and g(x) so that f( g(x) ) is that expression

We can do lots of things (there are lots of ways to choose f(x))
but let's do this one:
\[ f(x) = \frac{2}{x} + 3 \]

now what happens if we replace x with x^2 ? we would get
\[ y= \frac{2}{x^2} +3 \]
which is what we want.

we know (I hope) that if we use that f(x), then
f(g(x)) is
\[ f(g(x)) = \frac{2}{g(x)} + 3 \]

and if it turned out g(x)= x^2, we would have our answer.
ok, then let
g(x)= x^2

we have f(x) and g(x) so that f(g(x)) is what we want