Caroline Elliott

2022-01-22

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

kovaje5w

Beginner2022-01-23Added 4 answers

The functions are

$f\left(x\right)=\sqrt{{x}^{2}-2x}$

$g\left(x\right)=3x+4$

The composite function is

$f\left(g\left(x\right)\right)=f(3x+4)$

$=\sqrt{{(3x+4)}^{2}-2(3x+4)}$

$=\sqrt{9{x}^{2}+24x+16-6x-8}$

$=\sqrt{9{x}^{2}+18x+8}$

The domain is

$9{x}^{2}+18x+8\ge 0$

Therefore,

The solution to this quadratic equation is

$x=\frac{-18\pm \sqrt{{18}^{2}-4\cdot 9\cdot 8}}{18}$

$=\frac{-18\pm \sqrt{{18}^{2}-4\cdot 9\cdot 8}}{18}$

$=\frac{-18\pm \sqrt{36}}{18}$

$=\frac{-18\pm 6}{18}$

Therefore,

the composite function is undefined for

$(-\frac{24}{18},-\frac{12}{18})$

$=(-\frac{4}{3},-\frac{2}{3})$

The composite function is

The domain is

Therefore,

The solution to this quadratic equation is

Therefore,

the composite function is undefined for