Which Of The Following Is True About The Function Below

Understanding the characteristics of a function is fundamental in mathematics, allowing us to predict its behavior and apply it effectively in various fields. To determine which statements accurately describe a given function, we must analyze its properties, such as its domain, range, symmetry, and periodicity. Let's delve into a comprehensive exploration of how to assess and interpret these characteristics.

Analyzing Function Properties: A Step-by-Step Guide

To ascertain the truth about a function, a systematic approach is essential. Here's a step-by-step guide to help you dissect a function and understand its key properties.

1. Defining the Function

The first step involves clearly defining the function. A function is typically represented as f(x), where x is the input variable and f(x) is the output. Understanding the explicit formula or rule that defines the function is crucial. For example:

• f(x) = x² + 2x + 1
• g(x) = sin(x)
• h(x) = √(x)

2. Determining the Domain

The domain of a function is the set of all possible input values (x) for which the function is defined. Identifying the domain involves considering any restrictions on the input values. Common restrictions include:

• Division by zero: The denominator of a fraction cannot be zero. For example, in f(x) = 1/x, x cannot be 0.
• Square roots of negative numbers: In the real number system, you cannot take the square root of a negative number. For example, in g(x) = √x, x must be greater than or equal to 0.
• Logarithms of non-positive numbers: The argument of a logarithm must be positive. For example, in h(x) = log(x), x must be greater than 0.

To find the domain:

1. Identify potential restrictions based on the function's formula.
2. Express the domain as an interval or a set of intervals.

Example:

For f(x) = √(4 - x²), the domain is the set of all x such that 4 - x² ≥ 0. This implies x² ≤ 4, which means -2 ≤ x ≤ 2. Thus, the domain is [-2, 2].

3. Determining the Range

The range of a function is the set of all possible output values (f(x)) that the function can produce. Finding the range can be more challenging than finding the domain, and often involves analyzing the function's behavior. Some techniques include:

• Graphing the function: Visualizing the graph of the function can provide a clear picture of the range.
• Analyzing the function's behavior: Consider the function's minimum and maximum values, as well as its behavior as x approaches positive and negative infinity.
• Algebraic manipulation: Solve for x in terms of y (where y = f(x)). The domain of the resulting expression in terms of y is the range of the original function.

Example:

For f(x) = x², the range is all non-negative real numbers because any real number squared is non-negative. Thus, the range is [0, ∞).

4. Symmetry

Symmetry describes how a function behaves when its input is reflected across the y-axis or the origin. There are two main types of symmetry:

• Even functions: A function is even if f(-x) = f(x) for all x in its domain. Even functions are symmetric with respect to the y-axis. Example: f(x) = x².
• Odd functions: A function is odd if f(-x) = -f(x) for all x in its domain. Odd functions are symmetric with respect to the origin. Example: f(x) = x³.

To check for symmetry:

1. Replace x with -x in the function's formula.
2. Simplify the expression.
3. Compare the result with the original function.
   • If f(-x) = f(x), the function is even.
   • If f(-x) = -f(x), the function is odd.
   • If neither condition is met, the function is neither even nor odd.

Example:

For f(x) = x⁴ + 2x², we have f(-x) = (-x)⁴ + 2(-x)² = x⁴ + 2x² = f(x). Therefore, the function is even.

5. Periodicity

A function is periodic if its values repeat at regular intervals. That is, there exists a positive number P (the period) such that f(x + P) = f(x) for all x in the domain. Trigonometric functions like sine and cosine are classic examples of periodic functions.

To determine periodicity:

1. Check if the function's behavior repeats at regular intervals.
2. For trigonometric functions, identify the period by analyzing the function's formula. For example, the period of sin(ax) and cos(ax) is 2π/|a|.

Example:

For f(x) = sin(x), the period is 2π because sin(x + 2π) = sin(x) for all x.

6. Intercepts

Intercepts are the points where the function's graph intersects the x-axis and y-axis:

• x-intercepts: These are the points where f(x) = 0. To find x-intercepts, solve the equation f(x) = 0 for x.
• y-intercept: This is the point where x = 0. To find the y-intercept, evaluate f(0).

Example:

For f(x) = x² - 4, the x-intercepts are found by solving x² - 4 = 0, which gives x = ±2. The y-intercept is f(0) = 0² - 4 = -4.

7. Asymptotes

Asymptotes are lines that the function's graph approaches but never touches. There are three types of asymptotes:

• Vertical asymptotes: These occur where the function approaches infinity (or negative infinity) as x approaches a certain value. Vertical asymptotes typically occur at values where the function is undefined (e.g., where the denominator of a rational function is zero).
• Horizontal asymptotes: These describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, evaluate the limits lim x→∞ f(x) and lim x→-∞ f(x).
• Oblique (slant) asymptotes: These occur when the degree of the numerator of a rational function is one greater than the degree of the denominator. Oblique asymptotes can be found using polynomial long division.

Example:

For f(x) = 1/x, there is a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

8. Increasing and Decreasing Intervals

A function is increasing on an interval if its values increase as x increases. Conversely, a function is decreasing on an interval if its values decrease as x increases. To find increasing and decreasing intervals:

1. Find the function's derivative, f'(x).
2. Determine the critical points by solving f'(x) = 0 and identifying where f'(x) is undefined.
3. Create a sign chart for f'(x), indicating the intervals where f'(x) is positive (increasing) and negative (decreasing).

Example:

For f(x) = x², f'(x) = 2x. The critical point is x = 0. For x < 0, f'(x) < 0, so the function is decreasing. For x > 0, f'(x) > 0, so the function is increasing.

9. Concavity and Inflection Points

Concavity describes the curvature of the function's graph. A function is concave up if its graph is curved upward, and concave down if its graph is curved downward. An inflection point is a point where the concavity changes.

To analyze concavity:

1. Find the function's second derivative, f''(x).
2. Determine the possible inflection points by solving f''(x) = 0 and identifying where f''(x) is undefined.
3. Create a sign chart for f''(x), indicating the intervals where f''(x) is positive (concave up) and negative (concave down).

Example:

For f(x) = x³, f'(x) = 3x² and f''(x) = 6x. The possible inflection point is x = 0. For x < 0, f''(x) < 0, so the function is concave down. For x > 0, f''(x) > 0, so the function is concave up. Thus, there is an inflection point at x = 0.

Examples and Applications

Let's apply these concepts to a few example functions.

Example 1: f(x) = (x - 1) / (x + 2)

1. Domain: The denominator cannot be zero, so x ≠ -2. The domain is (-∞, -2) ∪ (-2, ∞).
2. Range: To find the range, let y = (x - 1) / (x + 2). Solving for x, we get x = (2y + 1) / (1 - y). The denominator cannot be zero, so y ≠ 1. The range is (-∞, 1) ∪ (1, ∞).
3. Symmetry: f(-x) = (-x - 1) / (-x + 2), which is neither f(x) nor -f(x). The function is neither even nor odd.
4. Intercepts:
   • x-intercept: f(x) = 0 when x - 1 = 0, so x = 1. The x-intercept is (1, 0).
   • y-intercept: f(0) = (0 - 1) / (0 + 2) = -1/2. The y-intercept is (0, -1/2).
5. Asymptotes:
   • Vertical asymptote: x = -2 (where the denominator is zero).
   • Horizontal asymptote: lim x→∞ (x - 1) / (x + 2) = 1. The horizontal asymptote is y = 1.

Example 2: f(x) = sin(2x)

1. Domain: All real numbers, (-∞, ∞).
2. Range: The sine function oscillates between -1 and 1, so the range is [-1, 1].
3. Symmetry: f(-x) = sin(2(-x)) = sin(-2x) = -sin(2x) = -f(x). The function is odd.
4. Periodicity: The period is 2π/2 = π.
5. Intercepts:
   • x-intercepts: sin(2x) = 0 when 2x = nπ for any integer n, so x = nπ/2.
   • y-intercept: f(0) = sin(20) = 0*. The y-intercept is (0, 0).

Example 3: f(x) = x³ - 3x² + 2x

1. Domain: All real numbers, (-∞, ∞).
2. Range: All real numbers, (-∞, ∞).
3. Symmetry: f(-x) = (-x)³ - 3(-x)² + 2(-x) = -x³ - 3x² - 2x, which is neither f(x) nor -f(x). The function is neither even nor odd.
4. Intercepts:
   • x-intercepts: f(x) = x³ - 3x² + 2x = x(x² - 3x + 2) = x(x - 1)(x - 2) = 0, so x = 0, 1, 2.
   • y-intercept: f(0) = 0³ - 3(0)² + 2(0) = 0.
5. Increasing and Decreasing Intervals:
   • f'(x) = 3x² - 6x + 2. Setting f'(x) = 0, we find x = (6 ± √(36 - 24)) / 6 = (3 ± √3) / 3. The critical points are approximately x ≈ 0.423 and x ≈ 1.577.
   • Analyzing the sign of f'(x):
   • For x < 0.423, f'(x) > 0 (increasing).
   • For 0.423 < x < 1.577, f'(x) < 0 (decreasing).
   • For x > 1.577, f'(x) > 0 (increasing).
6. Concavity:
   • f''(x) = 6x - 6. Setting f''(x) = 0, we find x = 1.
   • Analyzing the sign of f''(x):
   • For x < 1, f''(x) < 0 (concave down).
   • For x > 1, f''(x) > 0 (concave up).

These examples illustrate how to systematically analyze a function to determine its domain, range, symmetry, periodicity, intercepts, asymptotes, increasing and decreasing intervals, and concavity.

Common Mistakes to Avoid

When analyzing functions, it's easy to make mistakes. Here are some common pitfalls to avoid:

• Incorrectly determining the domain: Always carefully consider all possible restrictions on the input values.
• Assuming all functions have symmetry: Not all functions are even or odd.
• Miscalculating the range: The range can be more challenging to find than the domain, so take your time and use multiple techniques.
• Forgetting to check for asymptotes: Asymptotes can significantly affect the behavior of a function.
• Making errors in differentiation: Double-check your derivatives to ensure accuracy.
• Misinterpreting sign charts: Be careful when interpreting the signs of the first and second derivatives.

Practical Applications

Understanding function properties is not just an academic exercise. It has numerous practical applications in various fields:

• Physics: Describing motion, forces, and energy using functions.
• Engineering: Designing structures, circuits, and systems using mathematical models.
• Economics: Modeling supply and demand, growth, and market trends.
• Computer science: Developing algorithms, data structures, and software applications.
• Statistics: Analyzing data, making predictions, and drawing conclusions.

By mastering the art of function analysis, you'll gain a powerful tool for solving problems and understanding the world around you.

Conclusion

Analyzing the properties of a function is a crucial skill in mathematics and its applications. By systematically determining the domain, range, symmetry, periodicity, intercepts, asymptotes, increasing and decreasing intervals, and concavity, you can gain a deep understanding of the function's behavior. This knowledge enables you to make accurate predictions, solve complex problems, and apply functions effectively in various fields. Remember to avoid common mistakes and practice regularly to hone your skills. With dedication and a methodical approach, you can master the art of function analysis and unlock its full potential.