Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x+1}$$

Step-by-Step Solution

Answer

Domain: \(\mathbb{R}\), Range: \((0,\infty)\), Asymptote: \(y=0\), Increasing.

Step 1: Determine the Domain of the Function

The domain of a function is the set of all possible input values (usually x-values) that will output a real number when substituted into the function. Since the exponential function \(f(x)=e^{x+1}\) is defined for all real numbers, the domain is \(\mathbb{R}\).

Step 2: Identify the Range of the Function

The range of a function is the set of all possible output values. For the exponential function \(f(x)=e^{x+1}\), the output values are always positive, since \(e^x > 0\) for any real number \(x\). Therefore, the range of \(f(x)\) is \((0, \infty)\).

Step 3: Find the Horizontal Asymptote

Exponential functions like \(f(x) = e^{x+1}\) have a horizontal asymptote. As \(x\) approaches negative infinity, \(e^{x+1}\) approaches \(0\). Therefore, the horizontal asymptote is \(y = 0\).

Step 4: Determine if the Function is Increasing or Decreasing

To determine if \(f(x) = e^{x+1}\) is increasing or decreasing, consider the derivative \(f'(x)\). Since the derivative of \(e^x\) is \(e^x\), and \(e^{x+1} = e \cdot e^x\), it follows that the derivative \(f'(x) = e^{x+1}\) is always positive for all \(x\). Thus, \(f(x)\) is increasing on its domain.

Step 5: Sketch the Graph by Hand

Using the information from the previous steps, sketch the graph of \(f(x) = e^{x+1}\):- The graph is above the x-axis for all \(x\), reflecting a range of \((0, \infty)\).- It will have a smooth curve starting near the horizontal asymptote \(y=0\) and rise sharply as \(x\) increases.- The graph is increasing and has no intercepts on the x-axis.

Step 6: Verify with Calculator Graph

Use a calculator graphing tool to verify your sketch. Enter the function \(f(x) = e^{x+1}\) to confirm the characteristics such as the domain \(\mathbb{R}\), the range \((0, \infty)\), and the fact that the function is increasing with the horizontal asymptote at \(y=0\).

Key Concepts

Domain and Range

The domain of a function describes all possible inputs, represented as x-values, which you can safely substitute into the function. For exponential functions like \( f(x) = e^{x+1} \), every real number makes a valid input. Because of this, the domain of our function is all real numbers, written as \( \mathbb{R} \).

• Domain: All real numbers \( \mathbb{R} \)

The range of a function is the set of all possible outputs or y-values. With exponential functions, such as ours, the graph never touches or crosses the x-axis. This means the output is always a positive number. Thus, the range of the function \( f(x) = e^{x+1} \) begins just above zero and extends to infinity.

• Range: \( (0, \infty) \)

Asymptotes

Asymptotes are lines that a curve approaches but never actually touches. In exponential functions, there is commonly a horizontal asymptote. For \( f(x) = e^{x+1} \), as x approaches negative infinity, the output, or the y-value, gets closer and closer to zero but never actually reaches it.
This creates a horizontal asymptote at the line \( y = 0 \). The curve will hug this line closely but will not cross it. This is a common characteristic of many exponential functions. Understanding this helps in drawing the graph accurately by recognizing where the curve "levels off" or tends toward.

• Horizontal Asymptote: \( y = 0 \)

Increasing and Decreasing Functions

To determine if a function is increasing or decreasing, one method used is to find the derivative. The derivative gives us essential insights about the rate of change of the function. For our exponential function \( f(x) = e^{x+1} \), the derivative is \( f'(x) = e^{x+1} \), which is always positive.
Since the derivative is greater than zero for all x-values in the domain, it indicates that the function increases throughout its entire domain. This positive derivative shows that as you move from left to right along the graph, the value of the function consistently gets larger. Hence, the exponential function \( f(x) = e^{x+1} \) is a classic example of an increasing function.

• Function Behavior: Increasing