Problem 239 - Calculus Volume 1

Question

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. $$f(x)=e^{x}+2$$

Step-by-Step Solution

Answer

Domain: (-∞, ∞), Range: (2, ∞), Horizontal asymptote: y = 2.

Step 1: Understanding the Function

The function given is an exponential function: \[ f(x) = e^{x} + 2 \] This function is a transformation of the basic exponential function \( e^x \), which means the graph should be shifted vertically.

Step 2: Identify Vertical Shift

The function \( f(x) = e^{x} + 2 \) indicates a **vertical shift** upwards by 2 units from the parent function \( e^x \). This transformation affects the range and horizontal asymptote.

Step 3: Determine Domain

The domain of any exponential function is all real numbers, since you can substitute any real number into \( x \) without any restrictions. Therefore, the domain of \( f(x) = e^x + 2 \) is \[ (-\infty, \infty) \]

Step 4: Determine Range

Since the function is shifted up by 2 units, the smallest value \( e^x \) can take is 2 when \( e^x \) approaches zero. Thus, the range of \( f(x) = e^x + 2 \) is:\[ (2, \infty) \]

Step 5: Identify Horizontal Asymptote

In the original function \( e^x \), the horizontal asymptote is \( y = 0 \). With the vertical shift of +2, the new horizontal asymptote is now:\[ y = 2 \]

Step 6: Sketch the Graph

To sketch the graph of \( f(x) = e^x + 2 \):1. Start by sketching the graph of \( e^x \).2. Move the entire graph up by 2 units.3. Ensure the curve approaches but never touches the horizontal asymptote \( y = 2 \).4. Indicate the domain as all x-values and range beginning just above the horizontal asymptote at \( y = 2 \).

Key Concepts

Domain and Range

The domain of an exponential function, such as our function \(f(x) = e^x + 2\), includes all real numbers. This means you can plug any value of \(x\) into the function without restriction, and it will produce a corresponding \(y\) value.

• Domain: \((-\infty, \infty)\)

Let's discuss the range. The range represents all possible output values the function can produce. Since \(f(x) = e^x\) starts at 0 and increases without bound, any vertical shift will affect the minimum \(y\) value but not the overall structure of increasing values. In our function \(f(x) = e^x + 2\), the entire graph is shifted upwards by 2 units. Therefore, no matter what \(x\) value you choose, the lowest \(y\) value is 2.

• Range: \((2, \infty)\)

This means the function never dips below 2, but shoots up towards infinity as \(x\) increases.

Vertical Shift

A vertical shift occurs when the entire graph of a function moves up or down by a certain number of units. For the function \(f(x) = e^x + 2\), the graph of the basic exponential function \(e^x\) is shifted upwards by 2 units.

• This transformation is uniform across all x-values.
• The shape of the exponential curve remains unchanged but is repositioned vertically.

Such a vertical shift impacts the range and the position of the horizontal asymptote, because the whole graph, including its limits, is raised to a new position. However, it does not affect the domain since \(x\) values remain unrestricted.In real-world terms, you can think of this vertical shift as simply raising the entire baseline of the function. This property is useful in understanding how different factors (like constants added to a function) affect its graphical representation.

Horizontal Asymptote

A horizontal asymptote in a function is a horizontal line that the graph approaches as \(x\) goes to positive or negative infinity. In simpler terms, it's the \(y\) value that the outputs get closer to but never quite reach.For the base function \(e^x\), the horizontal asymptote is \(y = 0\). This means as \(x\) becomes very large or very small, \(e^x\) gets closer and closer to 0 but never actually reaches it.In the function \(f(x) = e^x + 2\), the vertical shift has moved this horizontal asymptote up by 2 units:

• Horizontal Asymptote: \(y = 2\)

This transformation means the function now approaches \(y = 2\) as \(x\) grows larger and larger, but it will never fall below this value. Recognizing horizontal asymptotes is essential for understanding the long-term behavior of exponential functions and predicting how they behave as the input is significantly increased.