Question

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=e^{x+1} $$

Short answer

The graph of the function \(y=e^{x+1}\) is a standard exponential graph shifted 1 unit to the left. The domain of the function is all real numbers \((-\infty, +\infty)\), and the range is all positive real numbers \((0, +\infty)\).

Step-by-step solution

Understanding the Function

The given function is \(y=e^{x+1}\), which is an exponential function. It is a transformation of the standard function \(y=e^x\) , which has been shifted 1 unit to the left due to the +1 inside the exponent.

Graphing the Function

To graph the function, you could create a table of values for x and y, where y is calculated by substituting x into the function equation. For example, if \(x = -2\), then \(y = e^{(-2+1)} = e^{-1}\). Repeat this for several more values of x to generate the graph. Alternatively, a graphing calculator could also be used to plot the function directly.

Identifying the Domain

The domain of the function is all real numbers, since you can put any number into the equation and get out a real number. This is represented as \( (-\infty, +\infty) \)

Post 4: Identifying the Range

For the range, an exponential function always yields values greater than 0, so the range of this function will be \((0, +\infty)\)

Key concepts

Domain and Range

The concept of "domain" and "range" is fundamental when dealing with functions like exponential functions. The domain refers to all the possible input values (x-values) that a function can accept without any restrictions. For exponential functions such as \(y=e^{x+1}\), you can input any real number for \(x\). This means the domain is all real numbers, represented in interval notation as \((-\infty, +\infty)\). You never face an issue with exponential functions having x-values that are undefined, unlike other functions where domain restrictions might exist, such as square root or log functions.

On the other hand, the range is concerned with the possible output values (y-values) that a function can produce. For an exponential function like \(y=e^{x+1}\), the output, which is the range, will only be positive numbers. This is because an exponent of \(e\) (or any other positive base) will never result in zero or negative numbers. Therefore, the range here is \((0, +\infty)\). Understanding these two components—domain as all real numbers and range as all positive numbers—is crucial for analyzing exponential functions.

Graphing Exponential Functions

Graphing exponential functions can seem daunting at first, but it's made easier by following a structured approach either using a table of values or leveraging technology like a graphing calculator. When graphing a function such as \(y=e^{x+1}\), you can start by picking several values for \(x\) and calculating the corresponding \(y\) values. This helps create individual points on your graph.

• For instance, when \(x = 0\), plug it into the function to get \(y = e^1 = e\).
• Repeat this for values like \(x = -2, -1, 1, 2\), finding \(y\) to build a clearer picture of the curve.

These computations can then be plotted on a Cartesian plane to generate the characteristic exponential curve. This curve will show a rapid increase from left to right, illustrating the nature of exponential growth.

Alternatively, employing a graphing calculator saves time by automatically plotting the function. Just enter the function in the calculator's graphing mode, allowing it to visualize the curve with greater precision and sometimes offering insights into points of interest like intercepts or asymptotic behavior.

Transformations of Functions

Transformations modify basic functions to shift, stretch, or flip them on a graph, providing insights into how the equation impacts its visual representation. For \(y=e^{x+1}\), the base function here is \(y=e^x\). The transformation seen by the \(+1\) in the exponent shifts the graph of \(y=e^x\) one unit to the left. Understanding basic transformations can help predict graph behavior before plotting.

• The horizontal shift occurs whenever there's an addition or subtraction within the exponent. Here, adding 1 means shifting left.
• If it were \(e^{x-1}\), the graph would shift one unit to the right instead.

This is just one kind of transformation. Other transformations might include:

• Vertical shifts, which adjust the \(y\)-position of the graph without affecting the \(x\),
• Reflections, flipping the graph over an axis,
• And vertical or horizontal stretching/compressing, making the graph narrower or wider.

Getting comfortable with these basic transformations allows for greater understanding and prediction of more complex function graphing tasks.