Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch a graph of the given exponential function. $$ f(x)=2^{\sqrt{x / 4}} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 2^{\sqrt{x/4}} \) is an exponential curve starting at (0, 1), increasing exponentially to the right.

Step 1: Analyze the function

The given function, \( f(x) = 2^{\sqrt{x/4}} \), is an exponential function. Here, the exponential base is 2, and the exponent is \( \sqrt{x/4} \). It's critical to recognize that this modifies the typical form of an exponential function because the exponent involves a square root.

Step 2: Determine the domain

The domain of \( f(x) = 2^{\sqrt{x/4}} \) is determined by the square root in the exponent. The expression under the square root, \( x/4 \), must be non-negative. Thus, \( x \geq 0 \). Therefore, the domain of the function is all non-negative real numbers \( [0, \infty) \).

Step 3: Calculate key points

Evaluate the function at several key points within the domain to get an idea of the graph's shape. For example:- For \( x = 0 \), \( f(x) = 2^{\sqrt{0/4}} = 2^0 = 1 \).- For \( x = 4 \), \( f(x) = 2^{\sqrt{4/4}} = 2^1 = 2 \).- For \( x = 16 \), \( f(x) = 2^{\sqrt{16/4}} = 2^2 = 4 \).Notice that as \( x \) increases, \( f(x) \) also increases.

Step 4: Determine the behavior as x approaches infinity

As \( x \to \infty \), the exponent \( \sqrt{x/4} \to \infty \). Therefore, the function \( f(x) = 2^{\sqrt{x/4}} \to \infty \). This means the graph increases without bound as \( x \) goes to infinity.

Step 5: Sketch the graph

Using the calculated key points and understanding the function's behavior:- Start at the point (0, 1) on the graph.- Plot other calculated points: (4, 2) and (16, 4).- Connect them with a smooth curve that illustrates exponential growth.- Ensure the graph is entirely right of the \( y \)-axis, starting at \( x = 0 \) since the function is not defined for negative \( x \).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells us the set of input values (i.e., the values of \( x \)) for which the function is defined. For the given exponential function \( f(x) = 2^{\sqrt{x/4}} \), the domain is influenced by two key factors:

• The expression \( \sqrt{x/4} \) must be defined, which means that \( x/4 \) must be non-negative.
• This leads to \( x \geq 0 \), because a square root can only be taken of non-negative numbers.

Thus, the domain of this function includes all non-negative real numbers, which is represented in interval notation as \( [0, \infty) \). This simply means that the function starts at zero and has valid output for all positive \( x \) values, extending to infinity.

Graph Sketching

Graphing an exponential function like \( f(x) = 2^{\sqrt{x/4}} \) involves identifying key characteristics and plotting them on a coordinate system. Start by calculating and plotting some important points to define the shape of the graph.

• At \( x = 0 \), \( f(x) = 2^{\sqrt{0/4}} = 2^0 = 1 \), so the graph begins at the point (0, 1).
• At \( x = 4 \), \( f(x) = 2^{\sqrt{4/4}} = 2^1 = 2 \), which gives us the point (4, 2).
• At \( x = 16 \), \( f(x) = 2^{\sqrt{16/4}} = 2^2 = 4 \), resulting in the point (16, 4).

Using these points, you can sketch the function's curve. It begins at \( x = 0 \) and only exists for positive \( x \) values. The function's exponential nature shows as it smoothly curves upwards, reflecting growth, and never crosses the y-axis. This helps visualize its increasing trend as described by the function.

Behavior as x Approaches Infinity

The behavior of a function as \( x \to \infty \) helps us understand how it changes far out in the positive direction. For the function \( f(x) = 2^{\sqrt{x/4}} \), the expression \( \sqrt{x/4} \) increases indefinitely as \( x \) gets larger.

• Since the exponent grows, \( f(x) \) continues to rise rapidly because any base greater than 1 raised to an increasing power results in a larger number.
• Consequently, \( f(x) \to \infty \) as \( x \to \infty \).

This indicates that the graph of the function will continue rising without bound, demonstrating the escalating growth typical of exponential functions. Through this understanding, we see that the function does not level off but climbs steadily upwards, forming a characteristic exponential shape as \( x \) extends further.