Let f(x)=log ₑ x and g(x)=(x⁴-2 x³+3 x²-2 x+2)/(2 x²-2 x+1). Then the domain of f ° g is

Solution

Maths sir · 4 steps

For log g(x) you only need g(x) > 0, so check the sign of top and bottom for every real x.

1. What to check. (f ° g)(x) = logₑ g(x) makes sense exactly when g(x) is defined and g(x) > 0. So the denominator must never be zero, and the whole fraction must be positive.
2. Denominator. 2x² - 2x + 1 = x² + (x-1)². Both squares cannot be zero at the same x, so the denominator is always positive and g is defined for every real x.
3. Numerator. Expanding gives (x² - x + 1)² = x⁴ - 2x³ + 3x² - 2x + 1, which is the numerator minus 1. So the numerator is a square plus 1.

   g(x) = ((x² - x + 1)² + 1)/(x² + (x-1)²) > 0 for all x ∈ ℝ

4. So. g(x) is positive for every real x, so logₑ g(x) is always defined. The domain of f ° g is ℝ, option (c).

The idea to keep. The domain of f ° g is the set of x where g(x) lands inside the domain of f. For a log, that means g(x) > 0. Try writing each polynomial as a sum of squares to check its sign quickly.

Answer from the NTA final answer key. Solution by PrepWiser.