91Ó°ÊÓ

When discussing exponential functions, like the one given by the equation \( y = e^{x+1} \), understanding the domain and range is key. The "domain" of a function refers to all the possible input values (in this case, \( x \) values) that the function can handle. For exponential functions, there are no restrictions on \( x \). This means you can substitute any real number into the equation. Therefore, the domain is all real numbers, written as \((-\infty, \infty)\).

On the other side, we have the "range," which represents all the possible output values (\( y \) values) the function can produce. Exponential functions of the form \( y = e^{x+1} \), will never yield negative values or zero because \( e \) raised to any power is always positive. This makes the range \( (0, \infty) \).

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

Identifying intercepts is crucial for graphing functions and understanding their behavior. For the function \( y = e^{x+1} \), let's find the intercepts.

The **y-intercept** is found by setting \( x = 0 \). Plug this into the equation to get \( y = e^{0+1} = e \). Thus, the y-intercept is the point \( (0, e) \), which you can plot on a graph. It marks where the graph of the function intersects the y-axis.

Exponential functions like \( y = e^{x+1} \) don't have an x-intercept because their graph never touches or crosses the x-axis. The value of \( y \) is always positive, no matter what value \( x \) takes.

• Y-intercept: \( (0, e) \)
• X-intercepts: None

An asymptote is a line that a graph approaches but never touches or crosses. For exponential functions like \( y = e^{x+1} \), identifying asymptotes helps in understanding the behavior of the graph at extreme values of \( x \).

In the case of \( y = e^{x+1} \), there is a horizontal asymptote. This occurs because as \( x \) goes to negative infinity, the value \( e^{x+1} \) gets closer and closer to zero but will never actually reach it. Therefore, the horizontal asymptote is the line \( y = 0 \).

Understanding where the asymptote lies gives insight into the potential growth or decay of the function. In this function, as \( x \) increases, the graph rises exponentially and will never approach the line \( y = 0 \) from above.

• Horizontal Asymptote: \( y = 0 \)