Let f (x) = log x and g(x) = 4 3 2 e x -2x +3x -2x+2 2 . Then the domain of f ∘ g is 2x -2x+1
Step-by-Step Solution
Key Concept: Apply the core result for domains, ranges and functional equations and simplify using the given constraints.
f (g(x)) = ln( x -2x +3x -2x+2 2 ) 2x -2x+1 (4) Since 2x - 2x + 1 > 0 2 \forallx \in R ∵ (-2) 2 - 4(2) < 0 Consider 4 3 2 x - 2x + 3x - 2x + 2 4 3 2 2 2 = (x - 2x + x ) + (x - 2x + 1) + (1 + x ) 2 2 2 2 = x (x - 1) + (x - 1) + (x + 1) > 0\forallx \in R \Rightarrow g(x) > 0\forallx \in R \Rightarrow ln f ((x)), f (x) > 0\forallx \in R \Rightarrow x \in R is domain
Correct Answer: 4