First slide

Introduction to real valued functions

Question

If $f (x) = \log_{[x - 1]} \frac{| x |}{x}$ , where [.] denotes the greatest integer function, then

Moderate

A

$D (f) = [3, \infty), R (f) = {0, 1}$
B

$D (f) = [3, \infty), R (f) = {0}$
C

$D (f) = (2, \infty), R (f) = {0, 1}$
D

$D (f) = (3, \infty), R (f) = {0}$

Solution

f(x) is defined for all x satisfying
$[x - 1] > 0, [x - 1] \neq 1$ and $\frac{| x |}{x} > 0$ .
Now, $[x - 1] > 0, [x - 1] \neq 1 \Rightarrow x \in [3, \infty)$
and , $\frac{| x |}{x} > 0 \Rightarrow x > 0$
$∴ D (f) = [3, \infty)$
Clearly, $\frac{| x |}{x} = 1$ for all $x \in [3, \infty)$
$∴ f (x) = \log_{[x - 1]} \frac{| x |}{x} = \log_{[x - 1]} 1 = 0$
So, $R (f) = {0}$

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