Let's analyze the given function:
$f(x) = 5 \left( \frac{1}{3} \right)^x$

1. Domain:

The domain of a function refers to all the possible input values (x-values). In this case, there are no restrictions on $x$, as the function is defined for all real numbers. Therefore, the domain is: $\boxed{(-\infty, \infty)}$

2. Range:

The range is the set of all possible output values (y-values). Since $\left( \frac{1}{3} \right)^x$ is an exponential decay function, it approaches 0 as $x \to \infty$ but never actually reaches 0. The constant multiplier of 5 shifts the function vertically. Thus, the function will output positive values but never reach zero. Hence, the range is: $\boxed{(0, \infty)}$

3. Horizontal Asymptote:

A horizontal asymptote is a horizontal line that the graph approaches but never touches as $x$ tends to positive or negative infinity. For exponential decay functions like this one, the horizontal asymptote occurs as $x \to \infty$. Since the function approaches but never reaches zero, the horizontal asymptote is: $\boxed{y = 0}$

4. y-intercept:

The y-intercept occurs where $x = 0$. Substituting $x = 0$ into the function: $f(0) = 5 \left( \frac{1}{3} \right)^0 = 5 \cdot 1 = 5$ Thus, the y-intercept is: $\boxed{(0, 5)}$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$
• Horizontal Asymptote: $y = 0$
• y-intercept: $(0, 5)$

Would you like to dive deeper into any of these concepts or ask further questions?

5 Related Questions:

1. How would the graph of this function look?
2. What happens if the base of the exponent were greater than 1?
3. How does changing the coefficient (like 5) affect the function's behavior?
4. What is the significance of the horizontal asymptote in real-world applications?
5. Can you find the x-intercept of this function?

Tip: Exponential functions often model decay or growth in natural processes such as population growth or radioactive decay. Understanding their behavior helps predict long-term trends.