Question

In Exercises \(1-4,\) graph the function. State its domain and range. $$y=3 \cdot e^{-x}-2$$

Short answer

The graph of the function \(y=3 \cdot e^{-x}-2\) is a decreasing exponential curve. The domain is all real numbers, i.e., \(-\infty < x < \infty\). The range is all real numbers less than 1, i.e., \(-\infty < y < 1\).

Step-by-step solution

Understand the nature of the function

The function \(y=3 \cdot e^{-x}-2\) is an exponential function where the base is \(e\) (Euler's number, approximately 2.71828). The '-x' in the exponent makes it a decreasing function, the '3' is a vertical stretch factor, and the '-2' is a vertical shift.

Determine the Domain of the function

The domain of this function is all real numbers. Since \(e^{-x}\) is defined for all real numbers \(x\), we can say that the domain is \(x: -\infty < x < \infty\) or \(-\infty < x < \infty\).

Determine the Range of the function

The range of an exponential function is all positive real numbers. But due to the '-2' vertical shift, it shifts the whole function two units below. Therefore, the range is \(y: -\infty < y < 1\).

Graph the function

To graph the function, create a table of values for \(x\) and \(y\). Computing the corresponding \(y\) values for the selected \(x\) values will produce the following table: | \(x\) | \(y\)| ------|------| -2 | 7.3 || -1 | 5 || 0 | 1 || 1 | -0.91 || 2 | -1.86 |Then plot these points on an XY graph, remembering that the graph should approach but never cross the horizontal asymptote \(y=-2\). The graph falls as it moves to the right, reflecting the function's decreasing nature.