函数hánshù

Core Concept

A function is one of the most fundamental concepts in mathematics, describing the relationship between two variables. Simply put, a function is an "input-output" mechanism: given an input value, according to a specific rule, we get a uniquely determined output value.

Mathematical Definition

Let $A$ and $B$ be two non-empty sets of numbers. If, according to a correspondence rule $f$, for every element $x$ in set $A$, there is a uniquely determined element $y$ in set $B$, then $f$ is called a function from $A$ to $B$, written as:

$y = f(x), \quad x \in A$

where:

• $x$ is the independent variable
• $y$ is the dependent variable
• $A$ is the domain
• The set of all values of $f(x)$ is called the range

Three Elements of a Function

A function is uniquely determined by three elements:

1. Domain - The set of all possible input values
2. Rule/Formula - The correspondence from input to output
3. Range - The set of all possible output values

Important: Two functions are equal if and only if their domains and correspondence rules are the same.

Common Function Types

1. Linear Function

$f(x) = kx + b \quad (k \neq 0)$ Graph: straight line

2. Quadratic Function

$f(x) = ax^2 + bx + c \quad (a \neq 0)$ Graph: parabola

3. Exponential Function

$f(x) = a^x \quad (a > 0, a \neq 1)$ Feature: rapid growth or decay

4. Logarithmic Function

$f(x) = \log_a x \quad (a > 0, a \neq 1)$ Feature: inverse of exponential function

5. Trigonometric Functions

$f(x) = \sin x, \cos x, \tan x, \text{etc.}$ Feature: periodic

Real-World Applications

Application 1: Speed and Time

Car traveling at 60 km/h. Relationship between time $t$ (hours) and distance $s$ (km)?

$s = f(t) = 60t$

Application 2: Profit Calculation

Product costs $50, sells for$80. Relationship between quantity $x$ and profit $P$?

$P = f(x) = 30x$

Application 3: Temperature Conversion

Celsius $C$ to Fahrenheit $F$:

$F = f(C) = \frac{9}{5}C + 32$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

Given function $f(x) = 2x + 1$, find $f(3)$.

Solution: $f(3) = 2(3) + 1 = 7$

Answer: 7

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Example 2: Intermediate (Difficulty ★★★☆☆)

Given $f(x + 1) = x^2 + 2x$, find $f(x)$.

Solution:

Let $t = x + 1$, then $x = t - 1$.

$f(t) = (t-1)^2 + 2(t-1) = t^2 - 1$

Therefore: $f(x) = x^2 - 1$

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Example 3: Advanced (Difficulty ★★★★☆)

Given $f(x) + 2f(\frac{1}{x}) = 3x$, find $f(x)$.

Solution:

From the equation: $f(x) + 2f(\frac{1}{x}) = 3x$ ... ①

Replace $x$ with $\frac{1}{x}$: $f(\frac{1}{x}) + 2f(x) = \frac{3}{x}$ ... ②

Solve: ①×2 - ②: $3f(\frac{1}{x}) = 6x - \frac{3}{x}$ $f(\frac{1}{x}) = 2x - \frac{1}{x}$

Replace $x$ with $\frac{1}{x}$: $f(x) = -x + \frac{2}{x}$

Common Mistakes

❌ Mistake 1: A function is just a formula

Correction: A function is a correspondence relationship, not necessarily expressed by a formula. It can be represented by tables, graphs, or descriptions.

❌ Mistake 2: One x can correspond to multiple y values

Correction: A function requires one input corresponds to exactly one output. If one $x$ gives multiple $y$ values, it's not a function.

Example: $x^2 + y^2 = 1$ (circle equation) is not a function.

❌ Mistake 3: Domain can be arbitrary

Correction: Domain must satisfy:

• Denominator ≠ 0
• Even roots ≥ 0
• Logarithm argument > 0
• Real-world constraints

Study Tips

1. ✅ Understand essence: Functions are relationships, not just formulas
2. ✅ Three elements: Domain, rule, and range are all essential
3. ✅ Classify types: Know characteristics of common functions
4. ✅ Analyze properties: Monotonicity, parity, periodicity are key
5. ✅ Real applications: Identify functional relationships in life

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💡 Exam Tip: Functions are core content in CSCA exams, with about 20% of questions directly or indirectly involving functions. Master them thoroughly!