The question asks for the domain and range of the function $f(x) = -\log_5(x + 2)$.

Domain:

• The logarithmic function $\log_b(x)$ is only defined for $x > 0$. For this particular function, $\log_5(x + 2)$, the argument $x + 2$ must be greater than 0.
  • So, $x + 2 > 0$ implies $x > -2$.
  • Therefore, the domain is $x > -2$.

Range:

• The function $f(x) = -\log_5(x + 2)$ involves a logarithmic expression. The logarithmic function typically has a range of all real numbers, but since there's a negative sign in front of the logarithm, it reflects the graph across the x-axis.
  • Thus, the range remains all real numbers.

Conclusion:

The correct answer is: A: Domain: $x > -2$, Range: all real numbers.

Would you like a detailed explanation or further clarifications?

Here are 5 related questions for further exploration:

1. How do you determine the domain of any logarithmic function?
2. What happens to the range of a function if it is multiplied by a negative number?
3. How would the domain and range change if the function was $\log(x - 3)$?
4. What is the effect of different logarithmic bases on the graph of a logarithmic function?
5. How does a horizontal or vertical shift affect the domain of logarithmic functions?

Tip: Remember, the domain of a logarithmic function is always determined by the condition that the argument of the logarithm must be strictly greater than zero.