To determine the domain and range of the exponential function $f(x) = 5^x + 5$ based on the graph:

1. Domain: The function $f(x) = 5^x + 5$ is defined for all real values of $x$ because there are no restrictions (such as division by zero or square roots of negative numbers). Therefore, the domain in interval notation is: $(-\infty, \infty)$

2. Range: Since $5^x$ is always positive for any real value of $x$ and grows exponentially as $x$ increases, the smallest value of $f(x) = 5^x + 5$ occurs as $x$ approaches negative infinity, where $5^x$ approaches zero, making the function approach $5$ from above. Thus, the range of the function is: $(5, \infty)$

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Related Questions:

1. What happens to the graph of $f(x) = 5^x + 5$ if you change the constant term to a different number?
2. How would the domain and range of $f(x) = 5^x$ compare to those of $f(x) = 5^x + 5$?
3. What are the horizontal and vertical asymptotes of the function $f(x) = 5^x + 5$?
4. How does the graph of $f(x) = 5^x + 5$ compare with the graph of $g(x) = e^x + 5$?
5. What effect does changing the base from 5 to another number have on the shape of the graph?

Tip:

When determining the range of an exponential function shifted vertically, always consider how the function behaves as $x$ approaches both large positive and negative values.