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Let's work through these two problems step by step.

### First Problem:
**Function:** \( f(x) = \frac{1}{7x + 49} \)

We are asked to find the domain of this function. The domain of a function is all the values of \(x\) for which the function is defined. Since the function is a fraction, the denominator must not be equal to zero. Therefore, we solve for \(x\) when the denominator is zero:

\[
7x + 49 = 0
\]

Solving for \(x\):

\[
7x = -49
\]
\[
x = -7
\]

So, \(x = -7\) is the only value that makes the denominator zero, meaning that \(x = -7\) is not in the domain of the function.

**Answer:** The only value not in the domain is \(x = -7\).

---

### Second Problem:
**Function:** \( f(x) = 4\sqrt{x} + 3 \)

We are asked to solve for \(x\) when \(f(x) = 8\). This gives us the equation:

\[
4\sqrt{x} + 3 = 8
\]

To solve for \(x\), we first subtract 3 from both sides:

\[
4\sqrt{x} = 5
\]

Next, divide both sides by 4:

\[
\sqrt{x} = \frac{5}{4}
\]

Now, square both sides to eliminate the square root:

\[
x = \left( \frac{5}{4} \right)^2 = \frac{25}{16}
\]

**Answer:** \(x = \frac{25}{16}\).

---

Do you want further details on any of these solutions or have any questions?

### Related Questions:
1. What is the general method to find the domain of a function with a fraction?
2. How do you solve for \(x\) when dealing with square roots in an equation?
3. Why does squaring both sides work when solving for a variable under a square root?
4. What are the conditions under which square roots are valid in a function?
5. How do you handle more complex fractional functions to determine their domains?

### Tip:
When solving square root equations, always check for extraneous solutions by plugging your answers back into the original equation.

Find the domain of the function f(x) = 1 / (7x + 49). What is the only value of x not in the domain?
Given f(x) = 4√x + 3, solve f(x) = 8.

Learn how to find the domain of a rational function and solve an equation involving a square root. Step-by-step solution for f(x) = 1 / (7x + 49) and f(x) = 4√x + 3, including isolating variables and solving for x.

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