Range of the function log 0-5 x 4-2 x 2-3A- - 0B- - 1C- -1-D- - 1-2-

The correct option is C ( ∞, 1]x^4 2x^2+3 nbsp;= x^4 2x^2+1+2 =(x^2 1)^2 + 2 Clearly, D(f) is ℝ Now, nbsp;2 ≤ (x^2 1)^2 + 2∞ As log_a x is strictly decr ...

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Standard XII

Mathematics

Range

Range of the ...

Question

Range of the function ${log}_{0.5} (x^{4} - 2 x^{2} + 3)$ is

(- \infty, 0)

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(- \infty, 1)

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(- \infty, - 1]

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(- \infty, \frac{1}{2}]

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Solution

The correct option is C $(- \infty, - 1]$
$x^{4} - 2 x^{2} + 3$
$= x^{4} - 2 x^{2} + 1 + 2$
$= (x^{2} - 1)^{2} + 2$
Clearly, $D (f)$ is $R$

Now, $2 \leq (x^{2} - 1)^{2} + 2 < \infty$
As ${log}_{a} x$ is strictly decreasing for $0 < a < 1$ ,
$- \infty < {log}_{1 / 2} (x^{4} - 2 x^{2} + 3) \leq {log}_{1 / 2} 2$
$\Rightarrow - \infty < {log}_{1 / 2} (x^{4} - 2 x^{2} + 3) \leq - 1$

Hence, $R (f) = (- \infty, - 1]$

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Range of the function log0.5(x4−2x2+3) is

The correct option is C (−∞,−1]x4−2x2+3 =x4−2x2+1+2 =(x2−1)2+2 Clearly, D(f) is R Now, 2≤(x2−1)2+2<∞ As logax is strictly decreasing for 0<a<1, −∞<log1/2(x4−2x2+3)≤log1/22 ⇒−∞<log1/2(x4−2x2+3)≤−1 Hence, R(f)=(−∞,−1]

Range of the function ${log}_{0.5} (x^{4} - 2 x^{2} + 3)$ is