Which Of The Following Is True About The Function Below

Understanding Function Analysis: Determining Which Statement is True

When analyzing mathematical functions, identifying which statements accurately describe their behavior is a fundamental skill. This process requires evaluating properties like continuity, differentiability, domain, range, and specific values. Let's explore how to systematically determine which statements about a given function are true, using a structured approach to ensure accuracy.

Steps to Analyze Function Properties

1. Identify the Function Definition
   Begin by clearly understanding the function's equation and any constraints. Functions may be defined algebraically, graphically, or as piecewise expressions. For example, consider:
   f(x) = { x² + 1, if x < 0 { 2x - 3, if x ≥ 0 }
   This piecewise function behaves differently based on the input value.

2. Evaluate Domain and Range
   The domain is all possible input values (x-values), while the range is all possible outputs (y-values). For the example above:

   • Domain: All real numbers (no restrictions).
   • Range: For x < 0, f(x) = x² + 1 ≥ 1. For x ≥ 0, f(x) = 2x - 3 ≥ -3. Thus, the range is [-3, ∞).

3. Check Continuity
   A function is continuous if it has no breaks, jumps, or holes. At x = 0:

   • Left-hand limit: lim(x→0⁻) f(x) = 0² + 1 = 1.
   • Right-hand limit: lim(x→0⁺) f(x) = 2(0) - 3 = -3.
     Since 1 ≠ -3, the function is discontinuous at x = 0.

4. Assess Differentiability
   Differentiability requires continuity and smoothness. Since the function is discontinuous at x = 0, it is not differentiable there. Elsewhere, each piece is differentiable:

   • For x < 0: f'(x) = 2x.
   • For x > 0: f'(x) = 2.

5. Calculate Specific Values
   Evaluate the function at key points:

   • f(0) = 2(0) - 3 = -3.
   • f(-1) = (-1)² + 1 = 2.
   • f(2) = 2(2) - 3 = 1.

6. Compare with Given Statements
   Test each statement against these properties. For instance:

   • Statement A: "f(x) is continuous for all x." → False (discontinuous at x=0).
   • Statement B: "The range includes all real numbers." → False (range is [-3, ∞)).
   • Statement C: "f(0) = 1." → False (f(0) = -3).
   • Statement D: "f(x) is differentiable at x = 1." → True (smooth and continuous at x=1).

Scientific Explanation of Function Behavior

Functions represent relationships between inputs and outputs, with properties rooted in calculus and algebra. Continuity ensures the function can be drawn without lifting a pen, requiring that:

• lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a).
  Differentiability further demands that the derivative (slope) exists, which implies continuity but not vice versa. Discontinuities occur when limits disagree or the function is undefined, as seen in our example at x=0.

The range is constrained by the function's behavior. Quadratic terms (e.g., x²) produce parabolas with minimum/maximum values, while linear terms extend infinitely. Combining these in piecewise functions creates hybrid ranges requiring analysis of each segment.

Common Properties and Analysis Techniques

Frequently Asked Questions

Q1: Can a function be differentiable but not continuous?
A1: No. Differentiability implies continuity. If a function isn't continuous at a point, it cannot be differentiable there.

Q2: How do piecewise functions affect domain/range analysis?
A2: Treat each piece separately. The domain is the union of all intervals, while the range combines outputs from each piece, considering overlaps and gaps.

Q3: What if a function has a "hole" (undefined point)?
A3: It is discontinuous at that point. For example, f(x) = (x² - 1)/(x - 1) is undefined at x=1, creating a hole despite simplifying to x + 1 elsewhere.

Q4: Are vertical asymptotes considered discontinuities?
A4: Yes. Functions like f(x) = 1/x are discontinuous at x=0 due to infinite limits and undefined behavior.

Conclusion

Determining which statements about a function are true requires a systematic evaluation of its core properties. By examining continuity, differentiability, domain, range, and specific values, you can confidently validate or refute each claim.

Extending the Toolkit: Monotonicity, Injectivity, and Composition Beyond continuity and differentiability, several other structural features are routinely examined when assessing the truth of assertions about a function.

Monotonicity. A function is said to be monotonically increasing on an interval if every rise in the input produces a non‑decreasing rise in the output; similarly, monotonically decreasing behavior is defined by a non‑increasing output. Detecting monotonicity is often achieved by inspecting the sign of the derivative: a positive derivative throughout an interval guarantees increasing behavior, while a negative derivative guarantees decreasing behavior. When the derivative changes sign, the function may attain local extrema, and the monotonicity claim must be qualified accordingly.

Injectivity and Surjectivity. An assertion that a function “maps distinct inputs to distinct outputs” is precisely the definition of injectivity (one‑to‑one). To verify injectivity, one can employ the horizontal‑line test on the graph or, algebraically, solve the equation f(x₁)=f(x₂) and show that it forces x₁=x₂. Conversely, surjectivity (onto) concerns whether every element of the codomain is attained as an output. For real‑valued functions defined on the entire real line, a common surjectivity test involves examining limits at ±∞ and any asymptotic behavior; if the function’s range is all real numbers, the claim of surjectivity holds. Composition of Functions. When several functions are combined, the truth of a statement about the composite often hinges on the properties of its constituent parts. If g maps a set A into B and f maps B into C, then f∘g is defined on A. The composite inherits continuity from its inner function provided the outer function is continuous at the relevant points, and differentiability propagates similarly under the chain rule. This interplay is especially valuable when evaluating piecewise definitions: the outer function may smooth out discontinuities introduced by the inner piece, or it may amplify them.

Inverse Functions. If a function is both injective and surjective (i.e., bijective) on a given domain and codomain, an inverse f⁻¹ exists. The existence of an inverse can be used to test statements such as “the inverse function is differentiable wherever the original function is differentiable and its derivative is non‑zero.” Proving this typically involves applying the inverse function theorem, which relates the derivative of f⁻¹ at a point to the reciprocal of f′ at the corresponding image point.

Periodicity. For functions that repeat values at regular intervals, statements about boundedness or symmetry often become simpler to assess. A function f is periodic with period T if f(x+T)=f(x) for all x in its domain. The smallest positive T is called the fundamental period. Periodicity can impose strong constraints on the range: a periodic function that is also continuous on a closed interval must attain a maximum and minimum, thereby bounding its range.

Putting It All Together

When confronting a set of assertions about a function, a disciplined workflow emerges:

1. Identify the domain and any hidden restrictions (e.g., division by zero, even‑root constraints).
2. Examine continuity at critical points — points where the definition changes, where denominators vanish, or where the function is explicitly undefined.
3. Check differentiability by verifying continuity and computing derivatives; remember that a derivative can fail to exist at cusps, corners, or vertical tangents.
4. Determine monotonicity via sign analysis of the derivative, which often informs injectivity.
5. Analyze injectivity and surjectivity using algebraic arguments or graphical tests.
6. Compute the range by collecting the outputs of each piece, paying special attention to limits at infinity and asymptotic behavior.