Question

1. $h(x) = \dfrac{1}{\sqrt4{x^2 - 5x}}$

Explanation:

Step1: Identify domain constraints

For the function $h(x) = \frac{1}{\sqrt[4]{x^2 - 5x}}$, the radicand of the 4th root must be positive (since we cannot divide by zero and even roots require non-negative inputs, and zero would make the denominator zero).
$$x^2 - 5x > 0$$

Step2: Factor the quadratic inequality

Factor the left-hand side to find critical points.
$$x(x - 5) > 0$$

Step3: Solve the inequality

The critical points are $x=0$ and $x=5$. Test intervals:

• For $x < 0$: $x$ is negative, $x-5$ is negative, product is positive (satisfies the inequality).
• For $0 < x < 5$: $x$ is positive, $x-5$ is negative, product is negative (does not satisfy).
• For $x > 5$: $x$ is positive, $x-5$ is positive, product is positive (satisfies the inequality).

Answer:

The domain of $h(x)$ is $(-\infty, 0) \cup (5, \infty)$