Скачать или смотреть CSCA Math Prep: Advanced Function Monotonicity Problem

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"For this question on the test, 90% of students got no points."

Welcome to another advanced math tutorial from Alifa Edtech. Today, we are tackling a notoriously difficult topic in high school mathematics: Monotonicity of Composite Functions (复合函数单调性).

► The Problem:

Find the monotonically increasing interval of the function: y = 1 / (4 + 3x - x^2)

This looks simple, but it requires a multi-step analysis that trips up almost everyone. You can't just find the derivative or look at the graph without breaking it down first.

► The Solution: A Systematic Approach

In this video, our expert teacher breaks down the solution into clear, logical steps.

Step 1: Find the Domain (Definition Domain). Before anything else, you must know where the function exists. The denominator cannot be zero.

4 + 3x - x^2 ≠ 0

Factorized: (x - 4)(x + 1) ≠ 0

So, x ≠ 4 and x ≠ -1.

Domain: (-∞, -1) ∪ (-1, 4) ∪ (4, +∞)

Step 2: Decompose the Function. We split the composite function into "inner" and "outer" layers:

Inner Function (u): u = 4 + 3x - x^2 (A downward-opening parabola)

Outer Function (y): y = 1 / u (An inverse proportion function)

Step 3: Analyse Monotonicity of Each Layer

Outer (y = 1/u): Always decreasing in its defined intervals.

Inner (u = -x^2 + 3x + 4):

Axis of Symmetry = -b / 2a = 3/2 (or 1.5).

Increasing: (-∞, 3/2]

Decreasing: [3/2, +∞)

Step 4: Apply the "Same Increase, Opposite Decrease" Rule. We want the whole function to be increasing. Since the Outer function is decreasing, we need the Inner function to be Decreasing (Opposite = Increase).

Where is the inner function decreasing? From [3/2, +∞).

Step 5: Combine with the Domain (The Final Trap!) We must intersect our finding [3/2, +∞) with the original Domain (-∞, -1) ∪ (-1, 4) ∪ (4, +∞).

The intersection is: [3/2, 4) and (4, +∞).

Final Answer: The monotonically increasing intervals are [3/2, 4) and (4, +∞).

► Why This Matters for the CSCA Exam

At Alifa Edtech, we champion Holistic Education. This problem isn't just about memorising a rule; it's about discipline. You must follow every step—Domain, Decomposition, Analysis, Combination—to get the right answer. This structured thinking is what top universities look for.

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Did you remember to check the domain first?

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