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.

When Are Two Functions the Same?

🌏 Cross-curriculum note: Chinese high school math is stricter about function equality than the conventions used in many other curricula. In some systems, two algebraically equivalent expressions are casually treated as "the same function." In the CSCA framework, the domain matters just as much as the formula.

Two functions $f$ and $g$ are considered identical only if both of the following hold:

1. Same domain — $f$ and $g$ are defined on exactly the same set of inputs.
2. Same correspondence rule — for every $x$ in the domain, $f(x) = g(x)$.

The variable name (whether it is $x$, $t$, or $u$) does not matter.

Worked Examples

Example A — Domain mismatch after simplification

$y = \frac{x - 1}{x^2 - 1} \quad \text{vs.} \quad y = \frac{1}{x + 1}$

The first has domain $\{x \mid x \neq \pm 1\}$; the second has domain $\{x \mid x \neq -1\}$. Even though they simplify to the same expression, they are not the same function because $x = 1$ is excluded from the first but allowed in the second.

Example B — Same function in disguise

$f(x) = |x| \quad \text{vs.} \quad g(x) = \sqrt{x^2}$

For every real $x$, $\sqrt{x^2} = |x|$. Both have domain $\mathbb{R}$ and produce the same output. These are the same function.

Example C — The $x^0$ trap

$f(x) = x^0 \quad \text{vs.} \quad g(x) = 1$

In Chinese high school, $0^0$ is undefined, so $f(x) = x^0$ has domain $\{x \mid x \neq 0\}$. But $g(x) = 1$ has domain $\mathbb{R}$. Not the same function.

Example D — Hidden absolute value

$f(x) = \sqrt{(x - a)^2} \quad \text{vs.} \quad g(x) = x - a$

$f(x) = |x - a|$, which only equals $x - a$ when $x \geq a$. The rules differ for $x < a$. Not the same function.

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

❌ Mistake 4: Treating algebraically equivalent expressions as the same function

Correction: This is the most common cross-curriculum confusion. Two functions are the same only when both their domain and rule match. Be especially careful with:

• $\frac{x-1}{x^2-1}$ vs. $\frac{1}{x+1}$ — simplification can shrink the domain
• $x^0$ vs. $1$ — $x^0$ excludes $x = 0$ in Chinese high school convention
• $\sqrt{x^2}$ vs. $x$ — the first equals $|x|$, not $x$

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!