Which Statement Is True About The Given Function

Diving into the world of functions, especially when presented with multiple statements about their characteristics, can feel like navigating a complex maze. The key is to break down each statement, understand the underlying principles of the function in question, and meticulously verify its truthfulness. Let's explore how to approach this, equipping you with the tools to analyze functions and determine which statement holds the key to accuracy.

Understanding the Function

Before dissecting statements, a solid understanding of the given function is paramount. This involves recognizing its type, domain, range, and any special properties it might possess.

• Type of Function: Is it linear, quadratic, exponential, trigonometric, logarithmic, or a more complex combination? Each type has unique characteristics that dictate its behavior.
• Domain: What are the permissible input values for the function? Are there any restrictions, such as division by zero or taking the square root of a negative number?
• Range: What are the possible output values of the function? Understanding the range helps in identifying maximum and minimum values, as well as any limitations on the function's output.
• Special Properties: Does the function exhibit symmetry (even or odd)? Is it periodic? Does it have asymptotes? These properties can significantly influence the truthfulness of statements made about the function.

To truly grasp the function, consider graphing it. Visualizing the function's behavior can provide valuable insights into its characteristics and help in verifying statements.

Common Types of Statements

Statements about a function can take various forms, each requiring a specific approach to verification. Here are some common types of statements:

• Increasing/Decreasing Intervals: These statements describe where the function's values are increasing or decreasing as the input variable increases.
• Concavity: This refers to the curvature of the function's graph. A function is concave up if its graph is shaped like a smile, and concave down if it's shaped like a frown.
• Extrema (Maxima/Minima): These are the maximum and minimum values of the function, either locally (within a specific interval) or globally (over the entire domain).
• Asymptotes: These are lines that the function approaches but never touches, either horizontally, vertically, or obliquely.
• Symmetry: This describes whether the function is symmetric about the y-axis (even function), the origin (odd function), or neither.
• Zeros/Roots: These are the values of the input variable that make the function equal to zero.
• Continuity: This refers to whether the function is continuous (no breaks or jumps) over a given interval.
• Differentiability: This refers to whether the function has a derivative at every point in a given interval.

Strategies for Verifying Statements

Now, let's delve into the strategies for verifying the truthfulness of statements about a function.

1. Increasing/Decreasing Intervals

To determine where a function is increasing or decreasing, we analyze its first derivative.

• Calculate the First Derivative: Find the derivative of the function, denoted as f'(x).
• Find Critical Points: Determine the values of x where f'(x) = 0 or f'(x) is undefined. These are the critical points of the function.
• Create a Sign Chart: Construct a number line and mark the critical points on it. Choose test values in each interval created by the critical points and evaluate f'(x) at those test values.
• Analyze the Sign Chart:
  • If f'(x) > 0 in an interval, the function is increasing in that interval.
  • If f'(x) < 0 in an interval, the function is decreasing in that interval.
  • If f'(x) = 0 at a critical point, the function may have a local maximum or minimum at that point.

Example:

Let's say we have the function f(x) = x^3 - 3x^2 + 2.

1. First Derivative: f'(x) = 3x^2 - 6x

2. Critical Points: 3x^2 - 6x = 0 => 3x(x - 2) = 0 => x = 0 or x = 2

3. Sign Chart:

   Interval	Test Value	f'(x)	Increasing/Decreasing
   x < 0	x = -1	9	Increasing
   0 < x < 2	x = 1	-3	Decreasing
   x > 2	x = 3	9	Increasing

Therefore, f(x) is increasing for x < 0 and x > 2, and decreasing for 0 < x < 2.

2. Concavity

To determine the concavity of a function, we analyze its second derivative.

• Calculate the Second Derivative: Find the second derivative of the function, denoted as f''(x).
• Find Possible Inflection Points: Determine the values of x where f''(x) = 0 or f''(x) is undefined. These are potential inflection points.
• Create a Sign Chart: Construct a number line and mark the possible inflection points on it. Choose test values in each interval created by the possible inflection points and evaluate f''(x) at those test values.
• Analyze the Sign Chart:
  • If f''(x) > 0 in an interval, the function is concave up in that interval.
  • If f''(x) < 0 in an interval, the function is concave down in that interval.
  • If f''(x) = 0 at a possible inflection point, and the concavity changes sign across that point, then it is an inflection point.

Example:

Using the same function f(x) = x^3 - 3x^2 + 2:

1. Second Derivative: f''(x) = 6x - 6

2. Possible Inflection Points: 6x - 6 = 0 => x = 1

3. Sign Chart:

   Interval	Test Value	f''(x)	Concavity
   x < 1	x = 0	-6	Concave Down
   x > 1	x = 2	6	Concave Up

Therefore, f(x) is concave down for x < 1 and concave up for x > 1. The point x = 1 is an inflection point.

3. Extrema (Maxima/Minima)

Extrema can be found using the first and second derivative tests.

• First Derivative Test:
  • Find the critical points of the function (where f'(x) = 0 or f'(x) is undefined).
  • Analyze the sign of f'(x) around each critical point:
    • If f'(x) changes from positive to negative at a critical point, then the function has a local maximum at that point.
    • If f'(x) changes from negative to positive at a critical point, then the function has a local minimum at that point.
    • If f'(x) does not change sign at a critical point, then the function has neither a local maximum nor a local minimum at that point.
• Second Derivative Test:
  • Find the critical points of the function (where f'(x) = 0 or f'(x) is undefined).
  • Evaluate the second derivative f''(x) at each critical point:
    • If f''(x) > 0, the function has a local minimum at that point.
    • If f''(x) < 0, the function has a local maximum at that point.
    • If f''(x) = 0, the test is inconclusive, and the first derivative test should be used.

Example:

Using the function f(x) = x^3 - 3x^2 + 2, we found critical points at x = 0 and x = 2.

• First Derivative Test (already performed in the increasing/decreasing section):
  • At x = 0, f'(x) changes from positive to negative, so f(x) has a local maximum at x = 0.
  • At x = 2, f'(x) changes from negative to positive, so f(x) has a local minimum at x = 2.
• Second Derivative Test:
  • f''(0) = -6 < 0, so f(x) has a local maximum at x = 0.
  • f''(2) = 6 > 0, so f(x) has a local minimum at x = 2.

4. Asymptotes

Asymptotes are lines that the function approaches but never touches.

• Vertical Asymptotes: Occur where the function is undefined, typically where the denominator of a rational function is equal to zero. Check the limit of the function as x approaches these values from both sides. If the limit goes to positive or negative infinity, then there is a vertical asymptote at that x-value.
• Horizontal Asymptotes: Determined by evaluating the limit of the function as x approaches positive and negative infinity. If the limit exists and is a finite number, then there is a horizontal asymptote at that y-value.
• Oblique (Slant) Asymptotes: Occur when the degree of the numerator of a rational function is exactly one greater than the degree of the denominator. To find the equation of the oblique asymptote, perform polynomial long division.

Example:

Consider the function f(x) = (x^2 + 1) / (x - 1).

• Vertical Asymptote: The denominator is zero when x = 1. Therefore, there is a vertical asymptote at x = 1.
• Horizontal Asymptote: The degree of the numerator (2) is greater than the degree of the denominator (1), so there is no horizontal asymptote.
• Oblique Asymptote: Performing polynomial long division, we get (x^2 + 1) / (x - 1) = x + 1 + 2/(x - 1). As x approaches infinity, the term 2/(x - 1) approaches zero. Therefore, the oblique asymptote is y = x + 1.

5. Symmetry

• Even Function: A function is even if f(-x) = f(x) for all x in its domain. Even functions are symmetric about the y-axis.
• Odd Function: A function is odd if f(-x) = -f(x) for all x in its domain. Odd functions are symmetric about the origin.

Example:

• f(x) = x^2 is an even function because f(-x) = (-x)^2 = x^2 = f(x).
• f(x) = x^3 is an odd function because f(-x) = (-x)^3 = -x^3 = -f(x).
• f(x) = x^2 + x is neither even nor odd.

6. Zeros/Roots

To find the zeros (or roots) of a function, set the function equal to zero and solve for x.

• f(x) = 0

Example:

Find the zeros of f(x) = x^2 - 4x + 3.

• x^2 - 4x + 3 = 0
• (x - 1)(x - 3) = 0
• x = 1 or x = 3

Therefore, the zeros of the function are x = 1 and x = 3.

7. Continuity

A function is continuous at a point if the following three conditions are met:

1. f(a) is defined (the function exists at that point).
2. The limit of f(x) as x approaches a exists.
3. The limit of f(x) as x approaches a is equal to f(a).

If any of these conditions are not met, the function is discontinuous at that point. Common discontinuities occur at points where the function has a vertical asymptote, a jump, or a hole.

8. Differentiability

A function is differentiable at a point if its derivative exists at that point. For a derivative to exist, the function must be continuous at that point, and the left-hand and right-hand limits of the difference quotient must be equal. Common non-differentiable points occur at sharp corners, cusps, or vertical tangents.

A Systematic Approach

When presented with multiple statements about a function, a systematic approach is crucial:

1. Understand the Function: Identify the type of function, its domain, and any special properties. Graphing the function is highly recommended.
2. Break Down Each Statement: Carefully analyze each statement and determine what it claims about the function.
3. Apply Verification Strategies: Use the appropriate techniques (derivatives, limits, etc.) to verify the truthfulness of each statement.
4. Eliminate False Statements: As you verify each statement, eliminate those that are false.
5. Identify the True Statement: The remaining statement(s) should be the true statement(s) about the function.
6. Double-Check: If possible, double-check your answer by using a different method or by graphing the function and visually inspecting its characteristics.

Example Scenario

Let's consider a function and several statements to illustrate the process:

Function: f(x) = x^3 - 6x^2 + 9x

Statements:

• A) f(x) is increasing on the interval (1, 3).
• B) f(x) has a local minimum at x = 1.
• C) f(x) is concave down on the interval (2, ∞).
• D) f(x) has a zero at x = -3.

Solution:

1. Understand the Function: f(x) is a cubic polynomial.

2. Break Down and Verify Each Statement:

   • A) f(x) is increasing on the interval (1, 3).

     • f'(x) = 3x^2 - 12x + 9

     • Critical points: 3x^2 - 12x + 9 = 0 => 3(x^2 - 4x + 3) = 0 => 3(x - 1)(x - 3) = 0 => x = 1, x = 3

     • Sign chart for f'(x):

       Interval	Test Value	f'(x)	Increasing/Decreasing
       x < 1	x = 0	9	Increasing
       1 < x < 3	x = 2	-3	Decreasing
       x > 3	x = 4	9	Increasing

     • Therefore, f(x) is decreasing on the interval (1, 3). Statement A is false.

   • B) f(x) has a local minimum at x = 1.

     • From the sign chart above, f'(x) changes from positive to negative at x = 1, indicating a local maximum, not a minimum. Statement B is false.

   • C) f(x) is concave down on the interval (2, ∞).

     • f''(x) = 6x - 12

     • Possible inflection point: 6x - 12 = 0 => x = 2

     • Sign chart for f''(x):

       Interval	Test Value	f''(x)	Concavity
       x < 2	x = 0	-12	Concave Down
       x > 2	x = 3	6	Concave Up

     • Therefore, f(x) is concave up on the interval (2, ∞). Statement C is false.

   • D) f(x) has a zero at x = -3.

     • f(-3) = (-3)^3 - 6(-3)^2 + 9(-3) = -27 - 54 - 27 = -108 ≠ 0. Statement D is false.

3. Identify the True Statement: In this case, none of the statements are true. This is perfectly possible, and it highlights the importance of rigorous verification.

Conclusion

Determining the truthfulness of statements about functions requires a solid understanding of fundamental concepts, a systematic approach, and careful application of verification techniques. By mastering these skills, you can confidently analyze functions and navigate the complexities of mathematical analysis. Remember to always double-check your work and be prepared for the possibility that none of the given statements are true!