Problem 243 - Calculus Volume 1

Question

Sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. \(f(x)=1-2^{-x}\)

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \((-\infty, 1)\); Horizontal Asymptote: \(y = 1\).

Step 1: Identify the Base Function

The given function is \(f(x) = 1 - 2^{-x}\). Here, the base function is \(g(x) = 2^{-x}\), which is an exponential function. The standard form of an exponential function is \(a^{x}\), where \(a > 0\). In this case, \(a = \frac{1}{2}\).

Step 2: Determine the Transformation

The transformation applied to \(g(x) = 2^{-x}\) is a vertical shift. The term \(1 - g(x)\) indicates that the graph of \(2^{-x}\) is reflected across the x-axis and then shifted up by 1 unit. The function \(f(x)\) is obtained by taking the inverted and shifted version of \(2^{-x}\).

Step 3: Sketch the Basic Graph

Start by sketching the graph of \(2^{-x}\), which decreases rapidly as \(x\) increases. Then reflect this graph across the x-axis and shift it up by 1 unit to obtain the graph for \(f(x) = 1 - 2^{-x}\).

Step 4: Determine the Domain

The domain of an exponential function like \(f(x) = 1 - 2^{-x}\) is all real numbers, written as \((-\infty, \infty)\). This is because you can substitute any real number for \(x\).

Step 5: Determine the Range

For the function \(f(x) = 1 - 2^{-x}\), as \(x\) approaches infinity, \(2^{-x}\) approaches zero, making \(f(x)\) approach 1. Conversely, as \(x\) approaches negative infinity, \(2^{-x}\) becomes very large, making \(1 - 2^{-x}\) deeply negative. Thus, the range is \((-\infty, 1)\).

Step 6: Identify the Horizontal Asymptote

The horizontal asymptote is the line that the function approaches as \(x\) approaches infinity or negative infinity. For \(f(x) = 1 - 2^{-x}\), the horizontal asymptote is \(y = 1\). As \(x\) goes to infinity, \(f(x)\) approaches but does not reach this value.

Key Concepts

Domain and Range

The domain of an exponential function refers to all the possible values that the variable, often denoted as \(x\), can take. For the function \(f(x) = 1 - 2^{-x}\), there are no limitations on \(x\). You can substitute any real number into the function, leading to a domain of \((-æfty, æfty)\).
In contrast, the range of the function describes all possible output values. To find the range for \(f(x) = 1 - 2^{-x}\), let's analyze the behavior of the function as \(x\) changes: - As \(x\) increases toward infinity, \(2^{-x}\) tends toward zero, causing \(f(x)\) to approach the value 1.
- As \(x\) decreases (tends toward negative infinity), \(2^{-x}\) grows significantly large, thus making \(1 - 2^{-x}\) a very negative number.
Combining these insights, the range of \(f(x)\) is \((-æfty, 1)\), meaning the outputs will cover all real numbers less than 1.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a function approaches as \(x\) heads towards infinity or negative infinity. It represents a value that the function will get arbitrarily close to but never quite reach. For \(f(x) = 1 - 2^{-x}\), the function approaches the value of 1 as \(x\) becomes very large (positive). This behavior is due to \(2^{-x}\) nearing zero, resulting in \(f(x) \approx 1\).
Therefore, the horizontal asymptote for this function is the line \(y = 1\). It's important to note that the graph will never actually cross this asymptote, serving instead as a "boundary" for where the function's outputs can approach but not surpass.

Transformations of Functions

Exponentials can be shifted and transformed in various ways, affecting their graphs. In the case of \(f(x) = 1 - 2^{-x}\), there are noticeable transformations applied to the base function \(g(x) = 2^{-x}\).
Here are the transformations to focus on:

• Reflection: The negative sign in \(-2^{-x}\) indicates a reflection across the x-axis, flipping the graph upside down.
• Vertical Shift: The "+1" part of the equation \(1 - 2^{-x}\) signifies a vertical shift upward by 1 unit. This raises the entire graph one unit higher on the coordinate plane.

To visualize these transformations, you could sketch the original \(2^{-x}\) like an upside-down bell curve, then reflect it and shift it up. Together, these transformations significantly shape the appearance and behavior of the graph.
Understanding transformations is crucial because it helps predict changes in the graph's position and style without recalculating each individual point.