Question

Find range of the function x^3-27÷x-3

Answer 1

F(x) = (x3-27)/ x-3

= (x- 3)^3/ (x-3 )

= (x-3 )^2

f{x} = (x-3)^2

(x- 3 ) ^2= x2-6x +9

Answer 2

Range of the function $y=\frac{x^3-27}{x-3}$

The Range is the set of all possible output values of the given function.

Using $a^3-b^3=(a-b)(a^2+ab+b^2)$

$\frac{x^3-27}{x-3}=\frac{(x-3)(x^2+3x+9)}{x-3}\\=x^2+3x+9$

The Simplified form of a given equation is a quadratic equation.

$y=x^2+3x+9$......(1)

From the equation, we can state that, $a=1, b=2$

since, $a > 0$ the given quadratic equation opens up the parabola curve.

Vertex of the curve,

$x=\frac{-b}{2a}$

substitute the value,

$x=\frac{-3}{2\times 1}\\x=\frac{-3}{2}\\x=-1.5$

substitute the value of $x$ in eq(1),

$y=(-1.5)^2+3\times (-1.5)+9\\y=2.25-4.5+9\\y=6.75$

vertex of the curve is $(-1.5,6.75)$

The range of the given equation is $[6.75,$ ∞$)$.