Question

number of integers in the domain of function f(x) = log[x²]^4-|x|+log2(√x) where [.] represent greatest integer function ​

Answer 1

EXPLANATION.

$\sf : \implies \: number \: of \: integer \: in \: the \: domain \: of \: function \\ \\ \sf : \implies \: f(x) = log_{ |x| {}^{2} }(4 - |x| ) + log_{2}( \sqrt{x} ) \\ \\ \sf : \implies \: x \: must \: be \: greater \: than \: 0 \: = x > 0 \\ \\ \sf : \implies \: \sqrt{x} > 0 \: \implies \: x \: \in \: (0, \infty ) \: \: \: .....(1)$

$\sf : \implies \: 4 - |x| > 0 \\ \\ \sf : \implies \: |x| < 4 \\ \\ \sf : \implies \: - 4 < x < 4 \\ \\ \sf : \implies \: x \: \in \: ( - 4,4) \: \: \: ......(2)$

$\sf : \implies \: |x| {}^{2} > 1 \\ \\ \sf : \implies \: x > 1 \: \: \: \: and \: \: \: \: x < 1 \\ \\ \sf : \implies \: x \: \in \: ( - \infty , - 1) \cup \: (1, \infty ) \: \: \: ......(3)$

$\sf : \implies \: from \: equation \: (1) \: \: and \: \: (2) \: \: and \: \: (3) \: \: we \: \: get \\ \\ \sf : \implies \: (1) \cap(2) \cap(3) \\ \\ \sf : \implies \: x \: \in \: (1,4) \: = \: answer$