Let's consider the function:

$f(x) = \frac{1}{\sqrt{25^2 - x^2}} = \frac{1}{\sqrt{625 - x^2}}$

Domain:

The domain consists of all $x$ values for which the function is defined. Since the square root is in the denominator, we need to ensure that:

1. $625 - x^2 \geq 0$ (the expression inside the square root must be non-negative).
2. $\sqrt{625 - x^2} \neq 0$ (the denominator must be non-zero).

Let's solve these conditions step by step.

1. $625 - x^2 \geq 0$:

$625 \geq x^2$ $x^2 \leq 625$ Taking square roots of both sides:

$-25 \leq x \leq 25$

2. $\sqrt{625 - x^2} \neq 0$:

$625 - x^2 \neq 0$ $x^2 \neq 625$ $x \neq 25 \quad \text{and} \quad x \neq -25$

Thus, the domain is:

$x \in (-25, 25)$

Range:

The range consists of all possible values of $f(x)$.

• The expression inside the square root, $\sqrt{625 - x^2}$, achieves its maximum value when $x = 0$, giving $\sqrt{625} = 25$.
• As $x$ approaches $\pm 25$, the expression inside the square root approaches zero.

Thus, the values of $f(x)$ range from:

$f(x) = \frac{1}{\sqrt{625 - x^2}}$

The minimum value occurs when $\sqrt{625 - x^2}$ is largest (i.e., when $x = 0$), giving $f(0) = \frac{1}{25}$. As $x$ approaches $\pm 25$, $f(x)$ becomes very large, approaching infinity.

Therefore, the range of the function is:

$f(x) \in (0, \frac{1}{25}]$

Summary:

• Domain: $(-25, 25)$
• Range: $(0, \frac{1}{25}]$

Would you like more details or have any questions about this?

Here are five related questions you can explore:

1. What is the domain of $\frac{1}{\sqrt{a^2 - x^2}}$ for any constant $a$?
2. How would the domain change if the function were $\frac{1}{25 - x^2}$?
3. What happens to the range if we change the square root to a cube root in the function?
4. How does the graph of $f(x) = \frac{1}{\sqrt{625 - x^2}}$ behave near the boundary points $x = \pm 25$?
5. Can we extend this function to complex numbers for values of $x$ outside the domain?

Tip: When finding the domain of a square root, remember to ensure the expression inside it is non-negative, as square roots of negative numbers are undefined in the real number system.