To analyze the function $f(x) = 4^x$, let’s go through the details step-by-step.

Part a: Domain and Range

1. Domain: The function $f(x) = 4^x$ is defined for all real values of $x$ because any real number $x$ can be used as an exponent for 4. Thus, the domain is: $(-\infty, \infty)$

2. Range: Since $4^x$ is an exponential function with a positive base greater than 1, it will always yield positive values. As $x \to -\infty$, $4^x \to 0$, but it never actually reaches 0. As $x \to \infty$, $4^x \to \infty$. Therefore, the range is: $(0, \infty)$

Part b: Asymptote

Since $f(x) = 4^x$ approaches 0 as $x \to -\infty$ but never actually reaches it, there is a horizontal asymptote at $y = 0$.

Final Answers

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

Would you like further details on exponential functions or how to determine asymptotes?

Additional Questions

1. What changes would occur to the domain and range if the function were $f(x) = 4^{-x}$?
2. How does the base of the exponential function (in this case, 4) affect the graph's growth rate?
3. What would happen to the asymptote if a constant were added to $f(x) = 4^x$?
4. How does the graph of $f(x) = 4^x$ compare to $f(x) = e^x$?
5. If the function were $f(x) = 4^{x+2}$, how would the graph shift?

Tip

For exponential functions of the form $f(x) = a^x$, where $a > 1$, the graph always increases as $x$ increases and approaches 0 as $x$ decreases. This pattern is consistent regardless of the base (as long as $a > 1$).