Question

jFind the domain of
$x + 7 \div {x}^{2} - 8x + 4$
​

Answer 1

Answer:

Step-by-step explanation:

$(f)x = y =\frac{x+7}{x^2-8x+4}$

For f(x) to be defined, denominator must not be 0,

hence,x²− 8x + 4 ≠ 0

⇒ (x−4)²− 16 + 4 ≠ 0

⇒ (x−4)²− 12 ≠ 0

⇒ (x−4)² ≠ 12

⇒ (x−4) ≠ ± 2√3

⇒  x ≠ 4 ± 2√3

Hence domain is (−∞,∞)−{4 ± 2√3}

Now

y=$\frac{x+7}{x^2- 8x+4}$

⇒yx² − 8xy + 4y = x+7

⇒yx²− x(8y+1) + 4y − 7 = 0

As x is real, hence

Discriminant  ≥ 0

⇒ (8y+1)²− 4 × y × (4y−7) ≥ 0

⇒64y²+ 1 + 16y − 16y²+28y ≥ 0

⇒ 48y² + 44y + 1 ≥ 0

Factorize using Shridharacharya's formula

$Y = \frac{-44\±\sqrt{44^2- 4*48*1} }{2*48}$

$=$ $\frac{-44\±\sqrt{1744} }{96}$

$= \frac{-44\±4\sqrt{1744} }{96}$

$= \frac{-11-\sqrt{109} }{24}$

$= {y - (\frac{-11-\sqrt{109} }{24}) \geq 0$

$\mathrm{Hence \ range \ is}$

$y \leq \frac{-11-\sqrt{ 109}}{24}$