Which Of The Following Is True About The Function Below

When faced with a question like "which of the following is true about the function below," the immediate challenge is that the function itself and the list of statements are missing. However, this presents a perfect opportunity to master the universal framework for analyzing any function. This article will equip you with a systematic, step-by-step methodology to deconstruct any function—whether polynomial, rational, exponential, or trigonometric—and accurately evaluate truth statements about its properties. You will learn to move beyond guesswork and build a confident, evidence-based analysis that works for algebra, calculus, and beyond.

The Universal Analysis Framework: Your Toolkit for Truth

To determine which statement about a function is true, you must first gather complete, unambiguous information about that function. Rushing to the options without this foundation is the most common source of error. Your analysis must be systematic, covering these core pillars:

1. Domain and Range: Where does the function exist? What values can it output?
2. Key Features: Intercepts (x and y), asymptotes (vertical, horizontal, slant), and holes.
3. Symmetry: Is it even, odd, or neither? This reveals graphical reflection properties.
4. Intervals of Increase/Decrease and Concavity: Where is the function rising, falling, or changing curvature?
5. Relative and Absolute Extrema: Local maxima/minima and global highs/lows.
6. End Behavior: What happens as x approaches positive and negative infinity?
7. Continuity and Differentiability: Are there any breaks, jumps, or sharp corners?

Let’s apply this framework to a classic, tricky example that often appears in such questions: f(x) = (x² - 4) / (x - 2).

Step-by-Step Analysis of f(x) = (x² - 4) / (x - 2)

1. Simplification and Domain: First, factor the numerator: x² - 4 = (x - 2)(x + 2). The function simplifies to f(x) = x + 2, but only where the original denominator is not zero. The original denominator, x - 2, is zero at x = 2. Therefore:

• Domain: All real numbers except x = 2. In interval notation: (-∞, 2) U (2, ∞).
• Critical Insight: The simplified form f(x) = x + 2 is a line, but the original function has a hole at (2, 4). It is not the same as the line g(x) = x + 2, which is defined everywhere.

2. Key Features:

• y-intercept: Set x=0. f(0) = (0-4)/(0-2) = (-4)/(-2) = 2. Point: (0, 2).
• x-intercept: Set numerator=0 (and denominator ≠0). (x-2)(x+2)=0 gives x=2 or x=-2. x=2 is excluded from the domain, so the only x-intercept is at x = -2. Point: (-2, 0).
• Vertical Asymptote: Occurs where denominator=0 and numerator≠0 after simplification. Here, at x=2, the factor (x-2) cancels completely. There is no vertical asymptote, only a removable discontinuity (hole) at x=2.
• Horizontal/Slant Asymptote: The simplified function is linear (degree 1). Since the degree of the numerator equals the degree of the denominator in the original unsimplified form, we compare leading coefficients. Both are 1, so the horizontal asymptote is y = 1. Wait, this seems to conflict with the simplified line y=x+2. This is a classic trap. The rules for asymptotes apply to the original function's form. For rational functions where degrees are equal, the horizontal asymptote is y = (leading coeff num)/(leading coeff den) = 1/1 = 1. However, because a factor cancels, the graph approaches y=1 only at the extremes, but the hole and the line's behavior create a more nuanced picture. Actually, for f(x) = (x²-4)/(x-2), after cancellation, the end behavior is dictated by the simplified linear term. The horizontal asymptote rule for equal degrees gives y=1, but since the simplified function is linear (degree 1 > 0), it has no horizontal asymptote; its end behavior is like a line. The correct analysis: the simplified function is f(x) = x+2 (with a hole), so as x→±∞, f(x)→±∞. There is no horizontal or slant asymptote.

3. Symmetry: Test f(-x): f(-x) = ((-x)² - 4)/((-x)-2) = (x² - 4)/(-x - 2) = -(x² - 4)/(x + 2). This is not equal to f(x) or -f(x). Therefore, the function is neither even nor odd.

4. Intervals of Increase/Decrease & Extrema: Using the simplified form f(x) = x + 2 (for x≠2), the derivative is f'(x) = 1. Since f'(x) = 1 > 0 for all x in