Which of the following is the graph of y -log 1-2 x -1-2-1-2log 24 x 2-4 x -1A-B-C-D-

The correct option is C This is not a standard expression whose graph we know, y=log_1/2 (x 1/2) + 1/2 log_2(4x^2 4x + 1) so,let's simplify the given expres ...

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Standard XI

Mathematics

Method of Intervals

which of the ...

Question

which of the following is the graph of $y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (4 x^{2} - 4 x + 1)$

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Solution

The correct option is C

This is not a standard expression whose graph we know,

$y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (4 x^{2} - 4 x + 1)$

so,let's simplify the given expression first.

$y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (4 x^{2} - 4 x + 1)$

= $l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (2 x - 1)^{2}$

= $l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} 4 {x - \frac{1}{2})^{2}}$

$y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} 4 + \frac{1}{2} l o g_{2} (x - \frac{1}{2})^{2}$

= - $l o g_{2} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} 4 + l o g_{2} (x - \frac{1}{2})^{\frac{2}{2}}$ ${l o g_{x}^{-} 1 y = - l o g_{x} y}$

$= - l o g_{2} (x - \frac{1}{2}) + l o g_{2} 2 + l o g_{2} (x - \frac{1}{2})$

$l o g_{2} k$ is defined only when k > 0

Above expression is defined only when $x - \frac{1}{2}$ > 0

Thus , y = $- l o g_{2} (x - \frac{1}{2}) + 1 + l o g_{2} (x - \frac{1}{2}) = 1$

so ,the graph of $y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (4 x^{2} - 4 x + 1)$

Domain of the graph is $x - \frac{1}{2}$ > 0

x > $\frac{1}{2}$

$x ϵ (\frac{1}{2}, \infty)$

Range of the graph is 1

$R ϵ {1}$

so,the graph is

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which of the following is the graph of y=log12 (x−12) + 12 log2(4x2 − 4x + 1)

which of the following is the graph of $y = l o g_{\frac{1}{2}} (x - \frac{1}{2}) + \frac{1}{2} l o g_{2} (4 x^{2} - 4 x + 1)$