- Recall that the exponential function \(e^x\) is always positive (\(e^x > 0\)) for any real input \(x\).
- Multiply by the positive scalar factor 4 to see that \(4e^x > 0\).
- Recognize that as \(x \to -\infty\), \(y \to 0\) and as \(x \to \infty\), \(y \to \infty\).
- The range is the set of all real numbers greater than \(0\).
- Final Answer: \(\boxed{\text{all real numbers greater than 0}}\)
Explanation
- Analysis of the exponential function \(e^x\)
The given function is \(y = 4e^x\). To find its range, we need to determine the set of all possible output values \(y\) as \(x\) varies over all real numbers. We start by considering the properties of the base exponential function \(f(x) = e^x\). The function \(e^x\) is defined for all real \(x\) (\(-\infty < x < \infty\)), and its value is always strictly positive, i.e., \(e^x > 0\) for all \(x\).
- Applying the transformation to find the range
The function \(y = 4e^x\) is a scaled version of the parent exponential function \(e^x\). Since \(e^x > 0\) for all real values of \(x\), we can multiply both sides of this inequality by 4. Because 4 is a positive constant, the direction of the inequality remains unchanged: \(4 \times e^x > 4 \times 0\), which simplifies to \(4e^x > 0\). Thus, \(y > 0\).
- Concluding the range of the function
As \(x\) approaches \(-\infty\), \(e^x\) approaches 0, so \(y = 4e^x\) approaches 0 but never reaches it. As \(x\) approaches \(\infty\), \(e^x\) increases without bound, so \(y = 4e^x\) also increases without bound. Therefore, the function \(y = 4e^x\) takes on all positive real values. The range of the function is all real numbers greater than 0. Comparing this with the given options, we identify the correct choice.
Examples
Understanding the range of exponential functions is crucial in various real-world models, such as population growth, radioactive decay, or interest accumulation. For instance, if you model the growth of a bacterial population over time \(t\) using the function \(P(t) = 4e^t\), knowing the range helps you realize that the population will always be greater than 0, regardless of the time passed, and it can grow infinitely large. Mastering these basic functions allows you to interpret such models effectively in biology, finance, and physics.