If $f (x) = [9^{x} - 3^{x} + 1], x \in (- \infty, 1)$ , then number of integers in the range of $f (x)$ is (where $[.]$ denotes greatest integer function)

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Solution

$9^{x} - 3^{x} + 1 = {(3^{x} - \frac{1}{2})}^{2} + \frac{3}{4} \geq \frac{3}{4}$
Now $\forall x \in (- \infty, 1)$
Range of $9^{x} - 3^{x} + 1$ is $[\frac{3}{4}, 7)$
$[9^{x} - 3^{x} + 1] \in 0, 1, 2, 3, 4, 5, 6$

Therefore, total number of integers in the range of $f (x)$ are $7$

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