Functions and rationalizing a radical in the denominator

[silent]

The problem is as listed:

f(x) = √x g(x) = 4x/x^2 - 7

I am supposed to find f(g(x))

When I start the first steps I get: √4x/x -√7 (Since the squared x in g(x) gets cancelled out by the radical it is just regular x right?)

I then cannot have the √7 in the bottom so I have to rationalize the radical so I multiply the fraction by x + √7 / x + √7


At this point I get so confused I dont know wher to start...

 

[pka]

The problem is as listed:
f(x) = √x g(x) = 4x/x^2 - 7
I am supposed to find f(g(x))...

If the question involves \(\displaystyle f(x)=\sqrt x~\&~g(x)=\dfrac{4x}{x^2-7}\) then \(\displaystyle f \circ g(x) = f(g(x)) = \sqrt {\frac{{4x}}{{{x^2} - 7}}} \)

But I don't know what you mean by " I have to rationalize the radical".

Can you clarify?

 

[silent]

Sorry if it seems like I dont make sense at all, i probably don't lol

but yes you are correct about what the problem is asking. What I tried to do is further reduce f(g(x)) and just totally messed up and got lost.

So basically my question is, is there anything I can do to further reduce it or can I just leave it underneath the radical?

 

[Quaid]

When I start the first steps I get: √4x/x -√7 (Since the squared x in g(x) gets cancelled out by the radical it is just regular x right?)

No, that's not right.

We do not simplify radicals like \(\displaystyle \sqrt{x^2-7}\) by taking square roots term-by-term to get \(\displaystyle x - \sqrt{7}\)


Do the instructions specifically tell you to rationalize the denominator for f(g(x)) ? If not, then the answer provided by pka should be good enough.

Cheers :cool:

 

[JeffM]

Sorry if it seems like I dont make sense at all, i probably don't lol

but yes you are correct about what the problem is asking. What I tried to do is further reduce f(g(x)) and just totally messed up and got lost.

So basically my question is, is there anything I can do to further reduce it or can I just leave it underneath the radical?

I believe (I think mrspi told me, but don't blame her if I am wrong) that the original purpose behind rationalizing a denominator was to simplify computation by hand.

That is, computing by hand \(\displaystyle \dfrac{7}{\sqrt{3}}\) involved messy division,

but computing by hand \(\displaystyle \dfrac{7\sqrt{3}}{3}\) does not involve messy division, just division by 3.

If that is true, there is not much practical purpose to rationalizing in an age of calculators and computers; it remains as merely a stylistic convention. Nor is there much purpose in rationalizing algebraic expressions that are not computable at all. Of course, as mmm says, it all depends on what your instructor or text tells you to do.