Caroline Elliott

2022-01-22

Function f and g are defined by $f\left(x\right)=\sqrt{\left({x}^{2}-2x\right)}$ and $g\left(x\right)=3x+4$. The composite function is undefined for $x\in \right]a;b\left[$ . Find the value of a and b?

kovaje5w

The functions are
$f\left(x\right)=\sqrt{{x}^{2}-2x}$
$g\left(x\right)=3x+4$
The composite function is
$f\left(g\left(x\right)\right)=f\left(3x+4\right)$
$=\sqrt{{\left(3x+4\right)}^{2}-2\left(3x+4\right)}$
$=\sqrt{9{x}^{2}+24x+16-6x-8}$
$=\sqrt{9{x}^{2}+18x+8}$
The domain is
$9{x}^{2}+18x+8\ge 0$
Therefore,
The solution to this quadratic equation is
$x=\frac{-18±\sqrt{{18}^{2}-4\cdot 9\cdot 8}}{18}$
$=\frac{-18±\sqrt{{18}^{2}-4\cdot 9\cdot 8}}{18}$
$=\frac{-18±\sqrt{36}}{18}$
$=\frac{-18±6}{18}$
Therefore,
the composite function is undefined for
$\left(-\frac{24}{18},-\frac{12}{18}\right)$
$=\left(-\frac{4}{3},-\frac{2}{3}\right)$

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