The given function is:

$y = \frac{1}{2} (4)^x$

1. Domain:

The domain of a function refers to the set of all possible input values (in this case, values for $x$) that will produce a valid output for the function.

For this function, there are no restrictions on $x$ because the expression $4^x$ is defined for all real numbers $x$. The exponential function $4^x$ can handle any real number input.

Domain:
$x \in (-\infty, \infty)$ or simply, all real numbers.

2. Range:

The range refers to the set of all possible output values (in this case, $y$) based on the domain.

The function $y = \frac{1}{2} (4)^x$ is an exponential function with base $4$, which is greater than 1. Exponential functions with positive bases greater than 1 (like $4^x$) grow without bound as $x \to \infty$, and approach zero as $x \to -\infty$. Since the function is multiplied by $\frac{1}{2}$, this affects the scaling but does not change the general growth or decay behavior.

As $x$ increases, $4^x$ grows larger, and thus $y = \frac{1}{2} (4)^x$ will also increase. As $x$ decreases, $4^x$ approaches zero, making $y$ approach zero as well. However, the function never actually reaches zero.

Range:
$y \in (0, \infty)$

Summary:

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

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Would you like more details or have any questions on this? Here are some related topics:

1. How does the base of an exponential function affect the domain and range?
2. What happens to the graph of $y = \frac{1}{2}(4)^x$ as $x \to \infty$?
3. How do transformations like multiplication and scaling affect the range of exponential functions?
4. What are the general properties of exponential functions with different bases?
5. How can logarithmic functions be used to find the inverse of exponential functions?

Tip: The base of an exponential function determines how quickly the function grows or decays. If the base is greater than 1, the function grows as $x$ increases. If the base is between 0 and 1, the function decays.