Let's examine the function $f(x) = 10^{0.2x} + 5$ to determine its domain and range.

1. Domain:

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined.

• The function $10^{0.2x}$ involves an exponential term, and the exponential function $10^y$ is defined for all real numbers $y$. Since $0.2x$ can be any real number as well, there are no restrictions on the values of $x$ that would make the function undefined.

Thus, the domain of the function is: $\boxed{(-\infty, \infty)}$

2. Range:

The range of a function refers to the set of all possible output values (y-values) the function can take.

• Consider the term $10^{0.2x}$. The exponential function $10^{0.2x}$ can take any positive real value, meaning that: $10^{0.2x} > 0 \text{ for all real } x.$
• Therefore, $f(x) = 10^{0.2x} + 5$ will always be greater than 5 since $10^{0.2x}$ is always positive. As $x \to -\infty$, $10^{0.2x}$ approaches 0, and as $x \to \infty$, $10^{0.2x}$ grows without bound.

Thus, the range of the function is: $\boxed{(5, \infty)}$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(5, \infty)$

Would you like further explanation or have any questions?

---

Here are some related questions to explore:

1. What happens to the graph of $f(x)$ as $x \to \infty$?
2. How would the domain and range change if the function were $f(x) = 10^{-0.2x} + 5$?
3. What is the value of $f(0)$?
4. How does the value of the base in the exponential affect the range?
5. What would happen if the constant $5$ was replaced with a negative number, like $-5$?

Tip: Exponential functions generally have a domain of $(-\infty, \infty)$, but their range can be affected by any constants added or subtracted.