For the function $f(x) = 3^x + 5$, let's determine the domain and range.

Domain

Since $3^x$ is defined for all real values of $x$, and adding 5 does not impose any restrictions, the domain of $f(x)$ is all real numbers.

Domain in interval notation: $(-\infty, \infty)$

Range

The function $f(x) = 3^x + 5$ represents an exponential function shifted upward by 5 units. The exponential function $3^x$ always produces positive values, so $3^x + 5$ will always be greater than 5. Therefore, the range of $f(x)$ is all values greater than 5.

Range in interval notation: $(5, \infty)$

Would you like further clarification on any part of this solution?

Here are 5 related questions for deeper understanding:

1. What happens to the range if the function changes to $f(x) = 3^x - 2$?
2. How would the domain and range change if the function was $f(x) = -3^x + 5$?
3. How does the graph of $f(x) = 3^x + 5$ compare to $f(x) = e^x + 5$?
4. What is the effect on the graph of $f(x) = 3^x$ if we add a negative constant, say $f(x) = 3^x - 3$?
5. How would you determine the asymptote of $f(x) = 3^x + 5$ using the graph?

Tip: For exponential functions like $a^x + c$, the constant $c$ represents the horizontal asymptote of the function.