- The function \(f(x) = (\frac{1}{4})^x\) is an exponential function with a base \(a = \frac{1}{4}\).
- Since \(0 < a < 1\), the output of the function is always positive for any real input.
- The behavior of the function confirms that it covers all values in the interval \((0, \infty)\).
- Thus, the statement "range of \(\{y \mid 0 < y < \infty\}\)" is correct.
- Final answer: \(\boxed{\text{True}}\)
Explanation
- Analyzing the Function and its Properties
The function given is \(f(x) = (\frac{1}{4})^x\). This is an exponential function of the form \(f(x) = a^x\), where \(a = \frac{1}{4}\). Because \(0 < a < 1\), the function is strictly decreasing. The domain of this function is all real numbers, meaning \(x \in (-\infty, \infty)\). To find the range, we observe the behavior of the output \(y\) as \(x\) varies. For any real \(x\), \(\left(\frac{1}{4}\right)^x\) is always positive. As \(x \to -\infty\), the function \(f(x)\) grows arbitrarily large (\(f(x) \to \infty\)), and as \(x \to \infty\), the function \(f(x)\) approaches \(0\) (\(f(x) \to 0\)). Therefore, the output \(y\) can take any positive value, covering the interval \((0, \infty)\).
- Evaluating the Statement
The statement provided is "range of \(\{y \mid 0 < y < \infty\}\)". We have determined that the range of the function \(f(x) = (\frac{1}{4})^x\) is indeed all real numbers \(y\) such that \(y > 0\). This set can be written in interval notation as \((0, \infty)\) or in set-builder notation as \(\{y \mid y > 0\}\), which is equivalent to \(\{y \mid 0 < y < \infty\}\). Therefore, the given statement correctly describes the range of the function \(f(x) = (\frac{1}{4})^x\).
Examples
Exponential functions are widely used in finance to model decay processes, such as the depreciation of a vehicle's value over time. If a car's value decreases according to a factor of \(1/4\) each year, the function \(f(t) = V_0(1/4)^t\) describes its value over time \(t\). Understanding the range of such a function tells us that the value will always remain above zero, even as it gets closer and closer to being worthless, which is a crucial insight for financial planning and asset valuation.