Problem 20 - Precalculus with Calculus Previews

Question

Determine the range of the given function. $$ f(x)=4-2^{-x} $$

Step-by-Step Solution

Answer

The range of the function is \((-\infty, 4)\).

Step 1: Understand the Function

The function given is \( f(x) = 4 - 2^{-x} \). This is a transformation of the exponential function \( g(x) = 2^{-x} \).

Step 2: Analyze the Behavior of the Exponential Component

The function \( g(x) = 2^{-x} \) is exponential. As \( x \to -\infty \), \( 2^{-x} \to \infty \), and as \( x \to \infty \), \( 2^{-x} \to 0 \). Thus, the range of \( 2^{-x} \) is \((0, \infty)\).

Step 3: Apply Transformation to the Function

The function transforms \( g(x) \) by reflecting it across the x-axis and then shifting it up by 4 units. As \( 2^{-x} \to 0 \), \( 4 - 2^{-x} \to 4 \). As \( 2^{-x} \to \infty \), \( 4 - 2^{-x} \to -\infty \).

Step 4: Determine the Range

Combining the transformations, \( f(x) \to 4 \) as the upper bound when \( x \to \infty \), and \( f(x) \to -\infty \) as the lower bound when \( x \to -\infty \). Therefore, the range of \( f(x) \) is \((-\infty, 4)\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are of the form \( a^x \), where \( a \) is a constant, and \( x \) is the variable. These functions exhibit rapid growth or decay and are fundamental in many scientific fields like biology, finance, and physics.

• **Growth:** When the base \( a > 1 \), the function is increasing, showing exponential growth.
• **Decay:** When \( 0 < a < 1 \), the function decreases, indicating exponential decay.

In the context of the original exercise, the function \( 2^{-x} \) is a classic example of exponential decay, where the base is less than \( 1 \) due to the negative exponent. As \( x \) increases, \( 2^{-x} \) approaches zero, which demonstrates its decaying nature.
Understanding how exponential functions behave is crucial for predicting their long-term behavior. These functions frequently appear in real-life situations like population growth, radioactive decay, and interest calculations. By grasping the nuances of exponential functions, you can better understand their transformations and ranges.

Function Transformations

Function transformations involve altering the original form of a function in various ways. These include translations, reflections, stretches, and compressions. Transformations help you understand how a function's graph can shift or change its orientation.

• **Translation:** Moving the graph horizontally or vertically without changing its shape.
• **Reflection:** Flipping the graph over a specific axis, changing its orientation.
• **Stretching/Compression:** Altering the graph's width or height.

For the function \( f(x) = 4 - 2^{-x} \), transformations are performed on \( 2^{-x} \):

• The subtraction by 2 implies a reflection over the x-axis.
• Adding 4 results in a vertical shift upwards.

The reflection causes the graph's direction to reverse, while the upward shift adjusts the entire graph higher on the coordinate plane.
By understanding these transformations, you can anticipate how they affect a function's range, domain, and graph. Exploring different transformations allows you to predict function behavior under various manipulations, pivotal for solving complex algebraic problems.

Asymptotic Behavior

Asymptotic behavior in functions describes how a function behaves as it approaches a certain line or point, known as an asymptote. This concept is essential for understanding limits and the behavior of graphs at infinity.
Types of asymptotes include:

• **Horizontal asymptotes:** The graph approaches a horizontal line as \( x \) or \( f(x) \) heads towards infinity.
• **Vertical asymptotes:** The graph approaches a vertical line, indicating the function's value becomes unbounded near that line.
• **Oblique asymptotes:** The graph approaches a slanted line, seen typically in rational functions.

In \( f(x) = 4 - 2^{-x} \), the function approaches a horizontal asymptote at \( y=4 \). As \( x \to \infty \), \( 2^{-x} \to 0 \), making \( f(x) \) approach 4 without ever reaching it. This creates a horizontal line that the function will never intersect, which is a clear example of asymptotic behavior.
Recognizing asymptotes helps in graphing functions properly and understanding their long-term tendencies. It is crucial for predicting a function's behavior during different intervals, aiding in calculus and advanced mathematical problem-solving.