First slide

Functions (XII)

Question

$Complete set of range of the function f (x) = \frac{1}{x^{6} + | x |^{3} - 1} is equal to$

Moderate

A

$[- 1, 0)$
B

$(- \infty, - 1] \cup (0, \infty)$
C

$(- \infty, - \sqrt{2}] \cup (0, \infty)$
D

$(- \infty, 0] \cup (1, \infty)$

Solution

$\begin{array}{l} Given that f (x) = \frac{1}{x^{6} + | x |^{3} - 1} \\ Clearly x^{6} + | x |^{3} \geq 0 \\ \Rightarrow x^{6} + | x |^{3} - 1 \geq - 1 \\ [ If x > a (a < 0) then \frac{1}{x} \in (- \infty, \frac{1}{a}) \cup (0, \infty)] \\ So range of f (x) = (- \infty, - 1] \cup (0, \infty) . \end{array}$
Therefore, the correct answer is (2).

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