Combining Functions: Given f(x)= √(-x2-3)-2, find g, h so f(x) = g(h(x))

[Hanna88]

[FONT=&quot]Hey can someone help me with this question?
Given f(x)f(x), find g(x)g(x) and h(x)h(x) such that f(x)=g(h(x))f(x)=g(h(x)) and neither g(x)g(x) nor h(x)h(x) is solely x.



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[FONT=&quot]

f(x)=−x2−3‾‾‾‾‾‾‾‾√−2​

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[Hanna88]

Combining Functions

HEY can someone help me with this question?
[FONT=&quot][FONT=&quot]Given[/FONT][FONT=&quot] f[/FONT][FONT=&quot]([/FONT][FONT=&quot]x[/FONT][FONT=&quot]), find g(x) and [/FONT][FONT=&quot]h[/FONT][FONT=&quot]([/FONT][FONT=&quot]x[/FONT][FONT=&quot]) such that [/FONT][FONT=&quot]f[/FONT][FONT=&quot]([/FONT][FONT=&quot]x[/FONT][FONT=&quot])= [/FONT][FONT=&quot]g[/FONT][FONT=&quot]([/FONT][FONT=&quot]h[/FONT][FONT=&quot]([/FONT][FONT=&quot]x[/FONT][FONT=&quot])) and neither g(x) nor h(x) is solely [/FONT][FONT=&quot]x[/FONT][FONT=&quot].
f(x)= √(-x²-3)-2[/FONT][/FONT]

 

[j-astron]

Hi Hanna88

What are you thinking with this one? What have you got so far? Show us your attempts to solve the problem, and we can tell you whether you're on the right track or not.

You need to express \(\displaystyle f(x)\) as a function of a function. I.e. you need to express it as \(\displaystyle g( h(x) )\). Can you think of a natural way to break the expression for \(\displaystyle f(x)\) up into two separate operations?

Remember that a function is like machine (a black box). You pass some input into the machine, and it spits out some output. So you pass \(\displaystyle x\) into the \(\displaystyle h\) function machine, and what comes out is \(\displaystyle h(x)\). Then you pass that result \(\displaystyle h(x)\) into the \(\displaystyle g\) function machine, and what comes out, \(\displaystyle g(h(x))\) is exactly equivalent to what you would have gotten if you had just passed \(\displaystyle x \) into the \(\displaystyle f \) function machine in the first place. I.e. \(\displaystyle f(x) = g(h(x))\). Using the \(\displaystyle f \) function machine is equivalent to using the \(\displaystyle h\) and \(\displaystyle g\) function machines one right after the other. This is what we mean when we say that \(\displaystyle f\) is a "composition" of two functions.

Side note: can you clarify what the mathematical expression for the function \(\displaystyle f(x)\) actually is? It's not clear from the way you typed it, and we can't solve the problem if we don't know this. From what you wrote, it looks like it might be:

\(\displaystyle \displaystyle f(x) = \sqrt{-x^2 -3} - 2\)

Is that correct?

 

[pka]

Hey can someone help me with this question?
Given f(x)f(x), find g(x)g(x) and h(x)h(x) such that f(x)=g(h(x))f(x)=g(h(x)) and neither g(x)g(x) nor h(x)h(x) is solely x.

f(x)=−x2−3‾‾‾‾‾‾‾‾√−2​

The above is un-readable.
But

HEY can someone help me with this question?
Given f(x), find g(x) and h(x) such that f(x)= g(h(x)) and neither g(x) nor h(x) is solely x.
f(x)= √(-x²-3)-2

is quite readable. The bad news is that there is no unique answer.

One answer might be \(\displaystyle h(x)=x^2+3~\&~g(x)=\sqrt{-x}-2\).

Can you find another pair?