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

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Answer 1 · 2018-07-05T16:45:45

The values of $a = - \frac{4}{3}$ $a=-4/3$ and $b = - \frac{2}{3}$ $b=-2/3$

The functions are

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

$g (x) = 3 x + 4$ $g(x)=3x+4$

The composite function is

$f (g (x)) = f (3 x + 4)$ $f(g(x))=f(3x+4)$

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

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

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

The domain is

$9 x^{2} + 18 x + 8 \geq 0$ $9x^2+18x+8>=0$

Therefore,

The solution to this quadratic equation is

$x = \frac{- 18 \pm \sqrt{18^{2} - 4 \cdot 9 \cdot 8}}{18}$ $x=(-18+-sqrt(18^2-4*9*8))/(18)$

$= \frac{- 18 \pm \sqrt{18^{2} - 4 \cdot 9 \cdot 8}}{18}$ $=(-18+-sqrt(18^2-4*9*8))/(18)$

$= \frac{- 18 \pm \sqrt{36}}{18}$ $=(-18+-sqrt36)/18$

$= \frac{- 18 \pm 6}{18}$ $=(-18+-6)/18$

Therefore,

the composite function is undefined for

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

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

graph{sqrt(9x^2+18x+8) [-9.227, 1.873, -2.24, 3.31]}

Function f and g are defined by f(x) = √(x2−2x)sqrt((x^2-2x)) and g(x) = 3x +4. The composite function is undefined for x ∈ ]a;b[ . Find the value of a and b?