91Ó°ÊÓ

In mathematics, understanding the domain and range of a function is crucial as they tell us what values the function can take. For the exponential function \(y = e^x\), the domain is quite straightforward. The domain includes all real numbers, which means that you can plug any real number into the function, and it will provide a meaningful result. This is expressed as \((-obreak ext{∞}, obreak ext{∞})\).

On the other hand, the range of the function \(y = e^x\) tells us the set of possible output values. For exponential functions like \(e^x\), the function will never produce zero or negative values. This is because an exponential function describes continuous growth for any real number \(x\). Thus, the range is \((0, obreak ext{∞})\). The graph of the function will always remain above the x-axis without touching it.

Graphing exponential functions such as \(y = e^x\) follows a unique pattern. Initially, the graph is very close to the x-axis for negative values of \(x\) but rises steeply for positive values of \(x\). This is due to the nature of exponential growth—when \(x\) is negative, \(e^x\) is a fraction; when \(x\) is positive, \(e^x\) increases rapidly.

To graph, start by plotting points. Using some calculated values can help. For example:

• When \(x = -2\), \(y \approx 0.135\)
• When \(x = -1\), \(y \approx 0.368\)
• When \(x = 0\), \(y = 1\)
• When \(x = 1\), \(y \approx 2.718\)
• When \(x = 2\), \(y \approx 7.389\)

After plotting these points, draw a smooth curve that reflects the exponential growth. The graph will demonstrate how quickly \(y = e^x\) increases as \(x\) becomes more positive.

Knowing the intercepts and asymptotes of the function \(y = e^x\) helps in visualizing the graph more accurately. **Intercepts** define where the graph crosses the axes. The **y-intercept** is found by setting \(x = 0\), which gives \(y = e^{0} = 1\). Thus, the graph crosses the y-axis at the point (0, 1).

Interestingly, there is no **x-intercept** for \(y = e^x\). Exponential functions like \(y = e^x\) will never touch or cross the x-axis, as they are never zero or negative.

**Asymptotes** give us a boundary the function approaches. For an exponential function \(y = e^x\), there is a horizontal asymptote at \(y = 0\). This means the graph comes infinitely close to the x-axis as \(x\) approaches negative infinity without actually touching it. Understanding these aspects can help in sketching an accurate graph.