It seems like you have multiple function expressions here that need to be analyzed or simplified. Let’s go through each one individually to clarify and address them properly.

1. Function $f(x) = 5\sqrt{x} - 3$

This function represents a square root function with a linear transformation:

• The term $\sqrt{x}$ suggests the function is defined for $x \geq 0$.
• The transformation $5\sqrt{x}$ scales the output by 5.
• Subtracting 3 shifts the function downward by 3 units.

2. Function $f(x) = \frac{8x + 1}{x^2 + 3x + 2}$

This is a rational function where:

• The numerator is $8x + 1$.
• The denominator is $x^2 + 3x + 2$.
• The domain is restricted where the denominator is not zero. To find where it’s undefined, set $x^2 + 3x + 2 = 0$, which factors as $(x + 1)(x + 2) = 0$. So, $x = -1$ and $x = -2$ are not in the domain.

3. Function $f(x) = \frac{8x + 1}{\sqrt{x^2 + 3x + 2}}$

This function is similar to the previous one but has a square root in the denominator:

• The square root restricts the function further since $x^2 + 3x + 2$ must be non-negative and cannot be zero.
• Since the expression under the square root factors as $(x + 1)(x + 2)$, the domain restrictions are $x \geq -2$ except for $x = -1$ and $x = -2$.

Summary of Important Points:

1. The first function $f(x) = 5\sqrt{x} - 3$ is defined for $x \geq 0$.
2. The second function $f(x) = \frac{8x + 1}{x^2 + 3x + 2}$ is undefined at $x = -1$ and $x = -2$.
3. The third function $f(x) = \frac{8x + 1}{\sqrt{x^2 + 3x + 2}}$ is defined for $x \geq -2$ but not at $x = -1$ and $x = -2$.

Would you like me to dive deeper into any of these functions or explore specific properties like asymptotes, intercepts, or graphs?

Related Questions:

1. How do you determine the domain of a function with a square root?
2. What are the asymptotes of the function $f(x) = \frac{8x + 1}{x^2 + 3x + 2}$?
3. How do you factor quadratic expressions like $x^2 + 3x + 2$?
4. What is the behavior of $f(x) = 5\sqrt{x} - 3$ as $x$ approaches infinity?
5. How do you simplify expressions involving square roots in the denominator?

Tip:

When analyzing the domain of a function, always look out for restrictions such as divisions by zero and square roots of negative numbers.