Sets, Relations & Functions
Domain of Composite Function
nta_pyq_2025_apr
Grade 11
Question:
Let $f(x) = \log_e x$ and $g(x) = \dfrac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}$. Then the domain of $f \circ g$ is:
$[0,\infty)$
$[1,\infty)$
$(0,\infty)$
$\mathbb{R}$
Step-by-Step Solution
Key Concept: Need $g(x) > 0$. Show both numerator and denominator are positive for all $x \in \mathbb{R}$.
$2x^2-2x+1>0$ always (discriminant $<0$). Numerator $= x^2(x-1)^2+(x-1)^2+(1+x^2)>0$ always. So $g(x)>0$ for all $x$, meaning $f(g(x))$ is defined for all $x\in\mathbb{R}$.
Correct Answer: $\mathbb{R}$