Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(h(x)=\sqrt{x^{2}+4}\)

Step-by-Step Solution

Answer

The function \( h(x) = \sqrt{x^2 + 4} \) is even.

Step 1: Understand the Concept of Even and Odd Functions

An even function satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain. An odd function satisfies \( f(-x) = -f(x) \). If neither condition holds, the function is neither even nor odd.

Step 2: Substitute \(-x\) into the Function

To determine if the function \( h(x) = \sqrt{x^2 + 4} \) is even or odd, substitute \(-x\) into the function: \[ h(-x) = \sqrt{(-x)^2 + 4} = \sqrt{x^2 + 4} \]

Step 3: Compare \( h(x) \) and \( h(-x) \)

The new expression \( h(-x) = \sqrt{x^2 + 4} \) is equal to the original function \( h(x) \). Therefore, the function satisfies the condition for even functions: \( h(x) = h(-x) \).

Step 4: Conclude Whether the Function is Even, Odd, or Neither

Since \( h(x) = h(-x) \), the function \( h(x) \) is even. There is no need to test for the odd function condition as the even condition is satisfied.

Step 5: Sketch the Graph of the Function

To sketch the graph of \( h(x) = \sqrt{x^2 + 4} \), consider the expression under the square root. The function resembles a horizontal translation of the basic function \( \sqrt{x^2} = |x| \), shifted upward by 4 units. The graph is symmetrical about the y-axis, representing even functions.

Key Concepts

Symmetry in Graphs

When we talk about symmetry in graphs, we are referring to the way graphs can be reflected or matched upon flipping across certain lines. A key aspect of symmetry is understanding what's meant by even and odd functions, as these relate directly to symmetry.

An **even function** is one where the graph is symmetrical about the y-axis. This means if you were to fold the graph along the y-axis, both halves would match perfectly. The mathematical rule for this is: If for every point \( (x, y) \) on the graph there is also a point \( (-x, y) \), then the function is even.

An **odd function** is one that has rotational symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it matches its original position. Odd functions follow the rule: If for every point \( (x, y) \) there is a corresponding point \( (-x, -y) \), then it is an odd function.

For the function \( h(x) = \sqrt{x^2 + 4} \), as shown in the solution, \( h(-x) = h(x) \), confirming that it’s an even function. This means its graph will show perfect symmetry across the y-axis.

Function Analysis

When analyzing a function, we look into its behavior, characteristics, and properties. Let's break it down to make it clearer and easier to understand.

• **Domain**: The domain of a function is the set of all possible input values \( x \). For \( h(x) = \sqrt{x^2 + 4} \), since you’re dealing with a square root that includes \( x^2 + 4 \), you can input any real value of \( x \). This is because \( x^2 + 4 \) is always positive.
• **Range**: The range is all possible output values. Here, since \( \sqrt{x^2 + 4} \) is always greater than or equal to 2 (due to the smallest value of \( x^2 + 4 \) being 4), the range starts from 2 upwards to infinity.
• **Continuity**: The function is continuous everywhere, which means there are no breaks, holes, or jumps in the graph.
• **Symmetry**: As previously established, the function is even, hence symmetrical about the y-axis.

This provides a comprehensive framework for understanding the function's characteristics, allowing better visualization and graphing later on.

Sketching Graphs

Sketching a graph of a function is an essential skill in understanding its behavior and characteristics. Let’s put these insights to use by sketching \( h(x) = \sqrt{x^2 + 4} \).

Start by recognizing that this function is similar to \( |x| \), which is a V-shaped graph. However, \( \sqrt{x^2 + 4} \) involves a horizontal and vertical transformation:

• **Vertical Shift**: The entire graph is lifted upwards, starting at y=2 (and not at the origin), since the minimum value of \( \sqrt{x^2 + 4} \) is 2.
• **Symmetry**: According to its even nature, the graph will be symmetrical about the y-axis.

To sketch:

• Draw the baseline starting at point (0, 2).
• Extend a smooth curve upwards symmetrically on either side of the y-axis, as both sides grow similarly because of the symmetry and even nature.
• Mark points for given x-values (like x = ±1, ±2) to better understand the rise and shape of the curve.

This graph gives a visual reassurance of the function’s properties, confirming its symmetry and range by the way it is spread vertically and horizontally.