First slide

Functions (XII)

Question

The domain of the function $f (x) = \frac{1}{\sqrt{x^{12} - x^{9} + x^{4} - x + 1}} is$

Moderate

A

$(- \infty, - 1)$
B

$(1, \infty)$
C

$(- 1, 1)$
D

$(- \infty, \infty)$

Solution

f (x) is defined for
$x^{12} - x^{9} + x^{4} - x + 1 > 0 \begin{array}{lr} \Rightarrow x^{4} (x^{8} + 1) - x (x^{8} + 1) + 1 > 0 \\ \Rightarrow (x^{8} + 1) x (x^{3} - 1) + 1 > 0 \end{array}$
$If x \geq 1 or x \leq - 1,$ then the above expression is positive.
$If - 1 < x \leq 0,$ the above inequality still holds.
$If 0 < x < 1, then x^{12} - x (x^{8} + 1) + (x^{4} + 1) > 0 [∵ x^{4} + 1 > x^{8} + 1 and so x^{4} + 1 > x (x^{8} + 1)]$
The domain of $f = (- \infty, \infty)$

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