To determine the domain of the function $f(x) = \frac{1}{\sqrt{8x+7}}$, we need to ensure the denominator is well-defined and non-zero. The conditions are:

1. The square root must be defined, meaning $8x + 7 \geq 0$.
2. The denominator cannot be zero, meaning $8x + 7 > 0$.

Step 1: Solve $8x + 7 > 0$

$8x > -7 \implies x > -\frac{7}{8}$

Conclusion

The domain of the function is $x > -\frac{7}{8}$. In interval notation, this is: $\boxed{(-\frac{7}{8}, \infty)}$

Would you like a more detailed explanation or have any questions about this?

Related Questions:

1. How do we determine the domain of a function with multiple conditions?
2. What happens to the domain if the numerator also had a variable?
3. How does the domain change if the denominator is a higher-degree polynomial?
4. Can the function be evaluated at the boundary $x = -\frac{7}{8}$?
5. How do square roots and fractions generally affect the domain?

Tip:

Always check both inequality conditions (non-negativity for square roots and non-zero for denominators) when determining domains involving fractions with roots.