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Find $g$

a) If $g∘(2f)=f+h$ where $f(x)=2x + 5$ and $h(x)=x^3 -2x$

b) If $(2f)∘g=f+h$ where $f(x)=\ln(x+2)$ and $h(x)=\sin(x^2)$

Thanks

1 Answers 1

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(a) Write out $f+h$ (that you can do directly). Then write out $2f$. Then try to expresss $f+h$ in terms of $2f$. Your expression is what $g$ does.

For example, if we have $f(x) = 3x+1$ and $h(x) = 36x^2+21x+3$, then $f+h = 36x^2 + 24x+4$. On the other hand, $2f = 6x+2$. Can we write $f+h$ in terms of $2f$? Yes: $(6x+2)^2 = 36x^2 + 24x + 4$, so $g(u) = u^2$ would satisfy $g\circ(2f)=f+h$.

(b) Similar to (a), but now you are doing $(2f)\circ g$ with the given $f$ and $h$.

In short: Write it out, figure it out.

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    @user7143: The reason I wrote all of that is that if I were trying to do this problem, I don't believe I would have every gotten to an equation of the form $(4x+10)^3 = x^3+5$, which is why I don't know how you got there. Hard to get someone unstuck if you have no inkling how they got stuck in the first place. – 2011-02-23