The function given is:

$f(x) = \sqrt{5x + 10} + 19$

This function is composed of two parts:

1. The square root function: $\sqrt{5x + 10}$
2. The constant term: $+19$

Domain of the Function

The expression under the square root, $5x + 10$, must be non-negative because the square root of a negative number is not defined in the real numbers. Therefore, we need:

$5x + 10 \geq 0$

Solving this inequality:

$5x \geq -10 \implies x \geq -2$

So, the domain of $f(x)$ is $x \geq -2$.

Range of the Function

• Since the square root function $\sqrt{5x + 10}$ outputs non-negative values, the smallest value it can take is 0 (when $x = -2$).
• Adding 19 to this non-negative value shifts the entire output up by 19.

Thus, the minimum value of $f(x)$ is $19$, and since the square root function can grow without bound as $x$ increases, $f(x)$ can take any value greater than or equal to 19. Therefore, the range of $f(x)$ is:

$\text{Range of } f(x) = [19, \infty)$

Summary

• Domain: $x \geq -2$
• Range: $[19, \infty)$

Would you like details or have any questions?

Here are some related questions to explore further:

1. What happens to the function if we modify it to $f(x) = \sqrt{5x + 10} - 19$?
2. How does the graph of $f(x)$ behave as $x$ increases?
3. What would the inverse function $f^{-1}(x)$ look like if it exists?
4. How does adding different constants to the square root function affect its range?
5. How would the domain and range change if the function were $f(x) = \sqrt{5x + 10} + 5$?

Tip: When analyzing functions involving square roots, always start by determining the domain based on the non-negativity requirement under the square root.