Let \( f(x) = \log x \) and \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] Then the domain of \( f \circ g \) is:
Step 1: Understanding domain constraints. The function \( f(x) = \log x \) requires \( x>0 \), so we must ensure \( g(x)>0 \) for \( f(g(x)) \) to be defined.
Step 2: Finding domain of \( g(x) \). Given: \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] The denominator is a quadratic equation: \[ 2x^2 - 2x + 1 \] Since the discriminant is negative, it is always positive.
Step 3: Solving for \( g(x)>0 \). Setting the numerator \( x^4 - 2x^3 + 3x^2 - 2x + 2>0 \), we find that \( x<0 \) satisfies this condition. Hence, \( g(x) \) is always positive. Thus, \( g(x)>0 \) for all \( x \), meaning the domain of \( f \circ g \) is \( \mathbb{R} \).
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]
\([A]\) (mol/L) | \(t_{1/2}\) (min) |
---|---|
0.100 | 200 |
0.025 | 100 |