Question

Determine the domain of the function

$$f(x)=\frac{1}{\sqrt{(x-2)^{2023}\,(x-3)^{2024}\,(x-4)^{2025}}}$$

Answer 1

$$(-\infty,\,2)\;\cup\;(4,\,\infty)$$

Answer 2

Step 1: Identify restrictions

• The expression under the square root must be positive (cannot be zero or negative).
• Exponents:
  • $(x-2)^{2023}$: odd exponent ⇒ sign of $(x-2)$ carries over.
  • $(x-3)^{2024}$: even exponent ⇒ always nonnegative, but zero at $x=3$ (excluded).
  • $(x-4)^{2025}$: odd exponent ⇒ sign of $(x-4)$ carries over.

Step 2: Exclude zeros of factors

• $x\neq2,3,4$ to avoid zero denominator.

Step 3: Positivity condition
We require

$$(x-2)^{2023}\,(x-4)^{2025} > 0$$

since $(x-3)^{2024}>0$ for all $x\neq3$.

Because both exponents are odd, this simplifies to

$$(x-2)\,(x-4) > 0.$$

Solve $(x-2)(x-4)>0$:

• Both factors positive $\implies x>4$.
• Both factors negative $\implies x<2$.

Step 4: Combine intervals
Exclude the points $2,3,4$. The domain is

$$(-\infty,2)\;\cup\;(4,\infty).$$