Question

section 4.5: function composition
score: 18/29 answered: 4/8
question 5
find two nontrivial functions ( f(x) ) and ( g(x) ) so ( f(g(x))=\frac{-7 sqrt{x+4}}{3} )
( f(x)=sqrt{x} )
( g(x)= )
question help: video message instructor

Explanation:

Step1: Analyze the given composite function

We have \(f(g(x))=\frac{-7\sqrt{x + 4}}{3}\). Given \(f(x)=\sqrt{x}\), we need to find \(g(x)\) such that when we substitute \(g(x)\) into \(f(x)\), we get the given composite function.
Let's assume \(y = g(x)\). Then \(f(y)=\sqrt{y}\). We want \(\sqrt{y}=\frac{-7\sqrt{x + 4}}{3}\). Squaring both sides (ignoring the negative sign for the non - trivial function construction as we are looking for the inner function structure), we consider the non - negative part of the right - hand side for the function composition analysis.

Step2: Solve for \(g(x)\)

If \(f(g(x))=\frac{-7\sqrt{x + 4}}{3}\) and \(f(x)=\sqrt{x}\), then we can rewrite \(f(g(x))\) as \(\sqrt{g(x)}\). So, \(\sqrt{g(x)}=\frac{-7\sqrt{x + 4}}{3}\). Squaring both sides (again, focusing on the function form), we get \(g(x)=\frac{49(x + 4)}{9}\). But if we consider the non - negative square root part (since \(f(x)=\sqrt{x}\) is defined for \(x\geq0\)), we can also think of it as \(g(x)=x + 4\) (because if \(f(x)=\frac{-7\sqrt{x}}{3}\) and \(g(x)=x + 4\), then \(f(g(x))=\frac{-7\sqrt{g(x)}}{3}=\frac{-7\sqrt{x + 4}}{3}\))

Answer:

\(f(x)=\frac{-7\sqrt{x}}{3}\), \(g(x)=x + 4\)