Which Of The Following Is True About The Function Below

Decoding the Function: A Comprehensive Analysis

This article delves deep into analyzing an unspecified function, exploring various aspects to determine which statements about it are true. Since no function is provided, we will create a hypothetical function and analyze it comprehensively, demonstrating the process you would use to analyze any given function. This process will cover crucial aspects like domain and range, behavior at critical points, and the impact of various parameters. We'll explore techniques applicable to various mathematical functions, including polynomial, trigonometric, exponential, and logarithmic functions. The examples will be applicable to a wide range of function types.

Let's assume our hypothetical function is:

f(x) = x³ - 3x² + 2x

We will now investigate several potential statements about this function and determine their truthfulness. This will serve as a template for analyzing any function you encounter.

H2: Domain and Range: The Foundation of Understanding

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce. Understanding the domain and range is crucial for grasping a function's behavior.

H3: Determining the Domain of f(x) = x³ - 3x² + 2x

Our example function, f(x) = x³ - 3x² + 2x, is a polynomial function. Polynomial functions are defined for all real numbers. Therefore, the domain of f(x) is all real numbers, which can be represented as (-∞, ∞). This means there are no restrictions on the input values; you can substitute any real number for x and get a valid output.

H3: Determining the Range of f(x) = x³ - 3x² + 2x

Determining the range of a polynomial function requires a bit more investigation. One approach is to analyze the function's behavior as x approaches positive and negative infinity. As x → ∞, f(x) → ∞, and as x → -∞, f(x) → -∞. This indicates that the function's output spans all real numbers.

However, for more complex functions, finding the range might require techniques like calculus (finding critical points and analyzing concavity). For our example, the range is also all real numbers, represented as (-∞, ∞).

H2: Critical Points: Unveiling the Function's Behavior

Critical points are points where the function's behavior changes significantly. These points can include local maxima, local minima, and inflection points. Analyzing critical points provides insights into the function's shape and overall behavior.

H3: Finding Critical Points using Calculus

To find the critical points, we need to use calculus. We first find the derivative of f(x):

f'(x) = 3x² - 6x + 2

Setting f'(x) = 0, we find the critical points:

3x² - 6x + 2 = 0

Solving this quadratic equation (using the quadratic formula), we find two critical points:

x = (6 ± √(36 - 24)) / 6 = (6 ± √12) / 6 = 1 ± √(1/3)

These are approximate values, and they represent potential locations of local maxima or minima.

H3: Determining the Nature of Critical Points (Maxima, Minima, or Inflection Points)

To determine whether these critical points are maxima or minima, we can use the second derivative test:

f''(x) = 6x - 6

Evaluating the second derivative at each critical point:

• For x = 1 + √(1/3): f''(x) > 0, indicating a local minimum.
• For x = 1 - √(1/3): f''(x) < 0, indicating a local maximum.

H2: Asymptotes and Limits: Exploring Function Behavior at Extremes

Asymptotes are lines that the function approaches but never touches. Limits describe the function's behavior as x approaches specific values (including infinity).

H3: Analyzing Asymptotes

Our example function, being a polynomial, has no asymptotes. Polynomial functions don't have horizontal or vertical asymptotes.

H3: Evaluating Limits

• Limit as x approaches infinity: lim (x→∞) f(x) = ∞
• Limit as x approaches negative infinity: lim (x→-∞) f(x) = -∞

These limits confirm our earlier observation about the range of the function.

H2: Intercepts: Where the Function Crosses the Axes

The x-intercepts are the points where the graph crosses the x-axis (where y = 0). The y-intercept is the point where the graph crosses the y-axis (where x = 0).

H3: Finding the x-intercepts

To find the x-intercepts, we set f(x) = 0:

x³ - 3x² + 2x = 0 x(x² - 3x + 2) = 0 x(x - 1)(x - 2) = 0

Therefore, the x-intercepts are at x = 0, x = 1, and x = 2.

H3: Finding the y-intercept

To find the y-intercept, we set x = 0:

f(0) = 0³ - 3(0)² + 2(0) = 0

Therefore, the y-intercept is at y = 0.

H2: Analyzing Potential Statements about the Function

Now, let's consider some potential statements about our function and determine their truthfulness based on our analysis:

• Statement 1: The function has three x-intercepts. TRUE (as shown above).
• Statement 2: The function has a local minimum and a local maximum. TRUE (confirmed by the second derivative test).
• Statement 3: The function has a horizontal asymptote. FALSE (polynomial functions do not have horizontal asymptotes).
• Statement 4: The domain of the function is all real numbers. TRUE (as explained in the domain section).
• Statement 5: The range of the function is all real numbers. TRUE (as concluded from the behavior at infinity).
• Statement 6: The function is always increasing. FALSE (it has a local maximum and minimum, indicating periods of increase and decrease).
• Statement 7: The y-intercept is at (0,0). TRUE (as calculated above).

H2: Conclusion: A Framework for Function Analysis

This comprehensive analysis of our hypothetical function demonstrates the systematic approach needed to determine the truthfulness of statements about any given function. The process involves understanding the function's domain and range, identifying critical points, analyzing asymptotes and limits, and determining intercepts. By applying these techniques – including calculus where necessary – we can accurately assess a function's characteristics and verify or refute claims about its behavior. Remember that the specific methods used will vary depending on the type of function being analyzed. This framework provides a robust foundation for understanding and interpreting mathematical functions. Remember to always adapt this approach to the specific function you are analyzing.