Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 is even. It is symmetric about the y-axis.

Step 1: Understand the Definitions of Even and Odd Functions

A function is **even** if for every x, \( f(-x) = f(x) \). It is **odd** if for every x, \( f(-x) = -f(x) \). To determine if a function is even or odd, check these conditions.

Step 2: Replace x with -x in h(x)

Given the function \( h(x) = \sqrt{x^2 + 4} \), replace \( x \) with \( -x \). \( h(-x) = \sqrt{(-x)^2 + 4} = \sqrt{x^2 + 4} \).

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

Compare \( h(x) = \sqrt{x^2 + 4} \) and \( h(-x) = \sqrt{x^2 + 4} \). Both expressions are identical, showing that \( h(x) = h(-x) \).

Step 4: Determine if the Function is Even, Odd, or Neither

Since \( h(x) = h(-x) \), the function \( h(x) \) is even. It is not odd because it does not meet the condition \( h(-x) = -h(x) \).

Step 5: Sketch the Graph

To sketch the graph of \( h(x) = \sqrt{x^2 + 4} \):1. Note that since \( x^2 \) and 4 are always positive, \( h(x) > 0 \) for all real \( x \).2. Plot key points, such as when \( x = 0 \), \( h(x) = \sqrt{0 + 4} = 2 \).3. Since \( h(x) \) is even, it is symmetric about the y-axis.4. The graph is a curve starting at \((0,2)\) ascending gradually on both sides.

Key Concepts

Function Symmetry

When investigating functions, symmetry plays an important role. A function can show two types of symmetry: even and odd. Understanding these can greatly simplify graph sketching and analyzing.

- An **even function** satisfies the condition: \( f(-x) = f(x) \) for every x in the domain. This means its graph will mirror itself about the y-axis. Think of even functions like a reflection in a pond.

- An **odd function** will satisfy: \( f(-x) = -f(x) \). In simple terms, this means its graph is symmetric with respect to the origin, giving it a rotating symmetry.

For the given function \( h(x) = \sqrt{x^2 + 4} \), we found that:

- \( h(-x) = \sqrt{(-x)^2 + 4} = \sqrt{x^2 + 4} \), meaning it’s identical to \( h(x) \). Therefore, it is an **even function**.

Recognizing these patterns not only helps in sketching but is also crucial in solving many function-related problems in calculus.

Graph Sketching

Graphing can visually bring a function to life, providing insights into its structure and behavior. Once the symmetry of the function is identified, as with our **even** function, the task becomes simpler.

For \( h(x) = \sqrt{x^2 + 4} \), here are some key steps to create a sketch:

• Start by noting that since the inner term \( x^2 \) is always non-negative, the smallest value for \( h(x) \) is its value at x = 0.
• At \( x = 0 \), \( h(0) = \sqrt{0 + 4} = 2 \). This point \( (0,2) \) is crucial for your graph.
• The even nature of the function means the graph will mirror along the y-axis. Plot key points on one side and reflect them across.
• The curve starts at \( (0,2) \) and rises smoothly away from the y-axis as x increases or decreases, forming a symmetric arch.

By carefully considering symmetry and key points, a reasonable graph sketch can serve shortcut to understanding function behavior.

Function Analysis

Analyzing a function involves understanding not just symmetry, but also its behavior over its entire domain. For \( h(x) = \sqrt{x^2 + 4} \), here’s a deeper look:

• **Domain**: The expression under the square root \( x^2 + 4 \) is always positive, meaning \( h(x) \) is defined for all real numbers \( x \).
• **Range**: Since the smallest value of \( \sqrt{x^2 + 4} \) is 2, the function only takes values \( y \geq 2 \).
• **Intercepts**: We've identified the y-intercept at \( (0,2) \). There are no x-intercepts because the minimum value is 2.
• **Behavior**: As \( x \) grows larger positively or negatively, the x-term grows rapidly. Thus, the function \( h(x) \) increases indefinitely.

These observations give a clearer picture of the function's properties. By examining both domain and range, intercepts, and general growth behavior, you build a solid understanding beyond merely graphing.