Question

Consider the real-valued function $f(x) = \frac{\log (3x-7)}{\sqrt{2x^2 - 7x + 6}}$. Find the domain of $f(x)$.

Accepted Answer

A. 1) Numerator $\log(3x-7)$ requires $3x - 7 > 0 \implies x > 7/3 \approx 2.33$. 2) Denominator $\sqrt{2x^2 - 7x + 6}$ requires $2x^2 - 7x + 6 > 0$. Roots of $2x^2 - 7x + 6 = 0$ are $x = 3/2$ and $x = 2$. So $2x^2 - 7x + 6 > 0$ for $x < 3/2$ or $x > 2$. Since $7/3 > 2$, for all $x > 7/3$, $2x^2 - 7x + 6$ is strictly positive. Thus, domain is $(7/3, \infty)$.

Domain of Real-Valued Function

Explanation

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