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Comprehensive Application of Function Properties

CSCA Comprehensive Application of Function Properties study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Comprehensive Application of Function Properties

1. Core Concepts

This section integrates all properties. Complex CSCA problems often combine **Monotonicity, Parity, Periodicity, and Symmetry**.

**Three Keys to Solving Problems**:

1. **Transformation**: Use Parity/Periodicity to move the problem from an unknown interval to a known interval (usually $[0, T]$ or $[0, +\infty)$).

2. **Decoding (Removing $f$)**: Use Monotonicity to strip away the function notation. E.g., $f(A) > f(B) \Rightarrow A > B$ (if increasing).

3. **Visualization**: Sketch a graph that satisfies the properties to visually determine roots or extrema.

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2. Common Models

#### (1) "Parity + Monotonicity" Model

* **Solving Inequalities**:

* If $f(x)$ is **Even** and **Increasing** on $[0, +\infty)$:

$$f(x) > f(a) \iff |x| > |a|$$

* If $f(x)$ is **Even** and **Decreasing** on $[0, +\infty)$:

$$f(x) > f(a) \iff |x| < |a|$$

* **Rule**: For Even functions, compare **Absolute Values**.

#### (2) "Periodicity + Parity" Model

* **Evaluation**: Use Periodicity to shift large inputs (e.g., $f(2025)$) to a small range, then use Parity to solve.

#### (3) "Symmetry + Periodicity" Model

* **Finding Period**:

* Two Axes $x=a, x=b \Rightarrow T = 2|a-b|$

* Two Centers $(a,0), (b,0) \Rightarrow T = 2|a-b|$

* One Axis $x=a$ + One Center $(b,0) \Rightarrow T = 4|a-b|$

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3. Typical Examples

**Example 1: Abstract Inequality**

Let $f(x)$ be an **Even** function on $\mathbb{R}$ and **Decreasing** on $[0, +\infty)$. Solve $f(2x-1) > f(\frac{1}{3})$.

**Solution**:

1. **Transform**: Since $f(x)$ is even, values depend on $|x|$.

2. **Decode**: Since decreasing on $[0, +\infty)$, "larger output means smaller absolute input".

$$f(2x-1) > f(\frac{1}{3}) \iff |2x-1| < |\frac{1}{3}|$$

3. **Calculate**:

$$-\frac{1}{3} < 2x-1 < \frac{1}{3}$$

$$\frac{2}{3} < 2x < \frac{4}{3} \Rightarrow \frac{1}{3} < x < \frac{2}{3}$$

**Example 2: Periodicity Calculation**

Let $f(x)$ be an **Odd** function on $\mathbb{R}$ with $f(x+2) = -f(x)$. For $0 \le x \le 1$, $f(x)=x$. Find $f(7.5)$.

**Solution**:

1. **Period**: $f(x+2)=-f(x) \Rightarrow T = 4$.

2. **Shift**: $f(7.5) = f(7.5 - 8) = f(-0.5)$.

3. **Parity**: $f(-0.5) = -f(0.5)$ (Odd function).

4. **Evaluate**: $f(0.5) = 0.5$ (Given range).

So $f(7.5) = -0.5$.

**Example 3: Parameter Range**

$f(x) = \frac{ax+1}{x+2}$ is increasing on $(-2, +\infty)$. Find the range of $a$.

**Solution**:

1. **Simplify**: $f(x) = a + \frac{1-2a}{x+2}$.

2. **Analyze**: This is a shifted $y = \frac{k}{x}$.

* $k < 0$ implies increasing.

3. **Solve**: $1-2a < 0 \Rightarrow a > \frac{1}{2}$.

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4. Common Pitfalls

* **Missing Absolute Value**: Solving $f(x_{even}) > f(a)$ as $x > a$ is wrong. It must be $|x| > |a|$ (if increasing).

* **Monotonicity Confusion**: Odd functions have the **SAME** monotonicity on symmetric intervals; Even functions have **OPPOSITE** monotonicity.

* **Period Formula**: $f(x+a)=-f(x)$ means $T=2a$, not $a$.