📚 IB Mathematics: Core Concepts and Problem-Solving Strategies for Function Properties | IB数学：函数性质的核心考点与解题方法
Functions are a central theme in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI). Understanding the properties of functions is not only essential for Paper 1 and Paper 2 questions, but also forms the foundation for calculus, statistics, and modelling. This article reviews the most tested concepts and provides systematic strategies for solving related problems.
函数是IB数学的核心主题，不论是在分析与方法（AA）还是应用与解释（AI）中，函数性质都占据重要地位。掌握函数的性质不仅是Paper 1和Paper 2解题的关键，更是学习微积分、统计与建模的基础。本文将系统梳理IB考试中最常考的函数性质概念，并给出对应的解题策略。
1. Domain and Range | 定义域与值域
The domain of a function is the set of all possible input values \(x\) for which the function is defined. The range is the set of all possible output values \(f(x)\). In IB exams, you must be able to find the domain and range from an equation, a graph, or a word problem.
函数的定义域是所有使函数有意义的输入值 \(x\) 的集合；值域是所有可能的输出值 \(f(x)\) 的集合。IB考试中，你需要能够从解析式、图像或实际问题中求定义域和值域。
- For polynomial functions, the domain is all real numbers, written as \(\mathbb{R}\).
- For rational functions, exclude values that make the denominator zero.
- For square root functions, the radicand must be greater than or equal to zero.
- For logarithmic functions, the argument must be strictly positive.
- 多项式函数的定义域为全体实数，记作 \(\mathbb{R}\)。
- 分式函数需排除使分母为零的取值。
- 根号函数要求被开方数大于或等于零。
- 对数函数要求真数严格大于零。
When finding the range, consider the horizontal extent of the graph. For a quadratic function \(f(x)=ax^2+bx+c\), if \(a>0\), the range is \([f_{\min}, \infty)\); if \(a<0\), the range is \((-\infty, f_{\max}]\).
求值域时，要观察图像在纵方向上的延伸范围。对于二次函数 \(f(x)=ax^2+bx+c\)，若 \(a>0\)，值域为 \([f_{\min}, \infty)\)；若 \(a<0\)，值域为 \((-\infty, f_{\max}]\)。
2. Composite and Inverse Functions | 复合函数与反函数
The composite function \((f \circ g)(x)=f(g(x))\) is obtained by applying \(g\) first and then \(f\). Its domain consists of all \(x\) in the domain of \(g\) such that \(g(x)\) lies in the domain of \(f\).
复合函数 \((f \circ g)(x)=f(g(x))\) 表示先施加 \(g\)，再施加 \(f\)。其定义域由满足“\(x\) 在 \(g\) 的定义域内，且 \(g(x)\) 在 \(f\) 的定义域内”的全部 \(x\) 构成。
For a function \(f\) to have an inverse, it must be one-to-one (injective). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y=x\). The domain of \(f^{-1}\) equals the range of \(f\), and vice versa.
函数 \(f\) 存在反函数的充要条件是它必须是一一对应的（单射）。函数 \(f^{-1}\) 的图像是 \(f\) 的图像关于直线 \(y=x\) 的对称图形。\(f^{-1}\) 的定义域等于 \(f\) 的值域，反之亦然。
\((f^{-1} \circ f)(x)=x\) and \((f \circ f^{-1})(x)=x\)
\((f^{-1} \circ f)(x)=x\) 且 \((f \circ f^{-1})(x)=x\)
- To find \(f^{-1}\), swap \(x\) and \(y\), then solve for \(y\).
- Always state the domain of the inverse, which is the range of the original function.
- 求反函数时，先将 \(x\) 与 \(y\) 互换，再解出 \(y\)。
- 务必写明反函数的定义域，即原函数的值域。
3. Even and Odd Functions | 函数的奇偶性
A function is even if \(f(-x)=f(x)\) for all \(x\) in its domain. Its graph is symmetric about the \(y\)-axis. A function is odd if \(f(-x)=-f(x)\) for all \(x\). Its graph is symmetric about the origin.
偶函数满足对定义域内任意 \(x\) 都有 \(f(-x)=f(x)\)，其图像关于 \(y\) 轴对称。奇函数满足 \(f(-x)=-f(x)\)，其图像关于原点对称。
| Property | Even function | Odd function |
| Condition | \(f(-x)=f(x)\) | \(f(-x)=-f(x)\) |
| Symmetry | About \(y\)-axis | About origin |
| Example | \(f(x)=x^2\), \(f(x)=\cos x\) | \(f(x)=x^3\), \(f(x)=\sin x\) |
To test for symmetry algebraically, replace \(x\) by \(-x\) and simplify. Many functions are neither even nor odd; in that case, simply state this with a counterexample.
判断奇偶性时，将 \(x\) 替换为 \(-x\) 并化简。许多函数既非偶函数也非奇函数；此时只需给出一个反例说明即可。
4. Periodicity | 周期性
A function is periodic with period \(p\) if \(f(x+p)=f(x)\) for all \(x\) in its domain. The smallest positive such \(p\) is called the fundamental period. Trigonometric functions are the most common periodic functions in IB.
若函数满足对定义域内任意 \(x\) 都有 \(f(x+p)=f(x)\)，则称其为周期函数，\(p\) 为周期。最小的正周期称为基本周期。三角函数是IB中最常见的周期函数。
\(\sin x\) has period \(2\pi\); \(\cos x\) has period \(2\pi\); \(\tan x\) has period \(\pi\)
\(\sin x\) 的周期为 \(2\pi\)；\(\cos x\) 的周期为 \(2\pi\)；\(\tan x\) 的周期为 \(\pi\)
For a transformed function \(f(x)=A\sin(Bx+C)+D\), the period is \(\frac{2\pi}{|B|}\). For tangent, the period is \(\frac{\pi}{|B|}\).
对于变换后的函数 \(f(x)=A\sin(Bx+C)+D\)，其周期为 \(\frac{2\pi}{|B|}\)。正切函数的周期为 \(\frac{\pi}{|B|}\)。
5. Monotonicity and Extrema | 单调性与极值
A function is increasing on an interval if \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\). It is decreasing if \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\). In IB, you often determine intervals of increase/decrease from a graph or using the first derivative.
若在某个区间内，\(x_1 < x_2\) 时有 \(f(x_1) < f(x_2)\)，则函数在该区间上单调递增。若 \(f(x_1) > f(x_2)\)，则单调递减。IB中常通过图像或一阶导数来判断增减区间。
A local maximum or minimum occurs at a critical point where \(f'(x)=0\) or \(f'(x)\) is undefined. Use the first derivative test or second derivative test to classify the point.
局部极大值或极小值出现在临界点，即 \(f'(x)=0\) 或 \(f'(x)\) 不存在的点。利用一阶导数符号变化或二阶导数符号来判断该点是极大值还是极小值。
| Test | Condition | Conclusion |
| First derivative | \(f’\) changes from + to − | Local maximum |
| First derivative | \(f’\) changes from − to + | Local minimum |
| Second derivative | \(f”(x) < 0\) | Local maximum |
| Second derivative | \(f”(x) > 0\) | Local minimum |
Remember to check endpoints when finding absolute extrema on a closed interval.
在闭区间上求最值时，不要忘记检查端点处的函数值。
6. Transformations of Functions | 函数的变换
IB questions frequently ask you to sketch or describe the transformation of a known function. The general rules are:
IB考试中常见要求是画出或描述已知函数的变换。基本规则如下：
- \(y=f(x)+a\): vertical translation by \(a\) units.
- \(y=f(x-a)\): horizontal translation by \(a\) units to the right.
- \(y=-f(x)\): reflection in the \(x\)-axis.
- \(y=f(-x)\): reflection in the \(y\)-axis.
- \(y=af(x)\): vertical stretch by factor \(a\).
- \(y=f(bx)\): horizontal compression by factor \(1/b\).
- \(y=f(x)+a\)：纵向平移 \(a\) 个单位。
- \(y=f(x-a)\)：向右平移 \(a\) 个单位。
- \(y=-f(x)\)：关于 \(x\) 轴对称。
- \(y=f(-x)\)：关于 \(y\) 轴对称。
- \(y=af(x)\)：纵向伸缩因子为 \(a\)。
- \(y=f(bx)\)：横向压缩因子为 \(1/b\)。
When multiple transformations are applied, pay attention to the order. For example, \(y=2f(x)+3\) means stretch vertically by 2, then translate up by 3. Horizontal transformations inside the argument follow a reverse order compared to what might seem intuitive.
当多个变换叠加时，要注意顺序。例如 \(y=2f(x)+3\) 表示先纵向拉伸2倍，再向上平移3个单位。内部的横向变换顺序往往与直觉相反，需要特别小心。
7. Asymptotes | 渐近线
Vertical asymptotes occur where the function approaches infinity as \(x\) approaches a finite value, typically where a rational function’s denominator is zero. Horizontal asymptotes describe the behaviour as \(x \to \pm \infty\).
当 \(x\) 趋近某个有限值而函数值趋于无穷大时，出现垂直渐近线，常见于分式函数分母为零处。水平渐近线描述 \(x \to \pm \infty\) 时的函数趋势。
For a rational function \(\frac{p(x)}{q(x)}\), compare the degrees of numerator and denominator:
对于有理函数 \(\frac{p(x)}{q(x)}\)，比较分子与分母的次数：
| Degree of numerator \(m\), denominator \(n\) | Horizontal asymptote |
| \(m < n\) | \(y=0\) |
| \(m = n\) | \(y=\frac{\text{leading coefficient of }p}{\text{leading coefficient of }q}\) |
| \(m > n\) | No horizontal asymptote (may have oblique) |
For logarithmic and exponential functions, remember the asymptotes: \(y=\ln x\) has a vertical asymptote at \(x=0\); \(y=e^x\) has a horizontal asymptote at \(y=0\).
对数和指数函数要记住其渐近线：\(y=\ln x\) 在 \(x=0\) 处有垂直渐近线；\(y=e^x\) 在 \(y=0\) 处有水平渐近线。
8. Key Features of Graphs | 函数图像的关键特征
When sketching a function, IB examiners expect to see all intercepts, turning points, asymptotes, and points of inflection clearly labelled. Understanding the relationship between a function and its derivatives is crucial.
画函数图像时，IB考官希望看到所有截距、极值点、渐近线和拐点都清晰标注。理解函数与其导数之间的关系至关重要。
The \(x\)-intercepts are solutions of \(f(x)=0\). The \(y\)-intercept is \(f(0)\). A point of inflection occurs where \(f”(x)=0\) and the concavity changes.
\(x\) 截距是方程 \(f(x)=0\) 的解，\(y\) 截距是 \(f(0)\)。拐点出现在 \(f”(x)=0\) 且凹凸性改变的位置。
If \(f”(x)>0\), the graph is concave up; if \(f”(x)<0\), the graph is concave down.
若 \(f”(x)>0\)，图像上凸（凹向上）；若 \(f”(x)<0\)，图像下凸（凹向下）。
Always use a sign diagram for the first derivative to determine the shape of the graph around critical points.
始终使用一阶导数符号表来确定临界点附近的图像形状。
9. Solving Equations and Inequalities | 求解方程与不等式
Function properties are often tested through equations like \(f(x)=g(x)\). The solutions correspond to the \(x\)-coordinates of intersection points of the two graphs. For inequalities such as \(f(x) < g(x)\), identify the intervals where the graph of \(f\) lies below the graph of \(g\).
函数性质常通过方程 \(f(x)=g(x)\) 来考查，其解对应两个图像交点的 \(x\) 坐标。对于不等式 \(f(x) < g(x)\)，需要找出 \(f\) 的图像位于 \(g\) 图像下方的区间。
When solving algebraically, always check for extraneous solutions, especially when squaring both sides or using logarithms.
代数求解时，务必检验增根，特别是在两边平方或使用对数时。
For piecewise functions, solve each branch separately and verify that solutions lie in the corresponding domain intervals.
对于分段函数，需对每一支分别求解，并验证解是否落在对应的定义域区间内。
10. Common Question Types and Strategies | 常见题型与解题策略
In Paper 1 (no calculator), you may be asked to determine the range of a quadratic, find the inverse of a simple function, or sketch a transformed graph. In Paper 2 (calculator allowed), you may need to solve transcendental equations or use technology to verify asymptotes and extrema.
Paper 1（不允许使用计算器）中，可能会要求你求二次函数的值域、求简单函数的反函数，或画出变换后的图像。Paper 2（允许使用计算器）中，可能需要解超越方程，或用科技工具验证渐近线和极值。
| Type | Strategy |
| Find domain | Set restrictions on denominator, radicand, logarithm argument |
| Find range | Use vertex, graph shape, or monotonicity on intervals |
| Inverse function | Swap \(x,y\), solve, state domain of inverse |
| Even/odd check | Evaluate \(f(-x)\) and compare with \(f(x)\) and \(-f(x)\) |
| Graph transformation | Apply one transformation at a time, label key points |
For longer questions, read the full problem carefully. Often the first part asks for a graph or a range, and later parts require you to use that information to solve an equation or inequality.
对于较长的题目，请仔细阅读完整问题。通常第一部分要求画图或求值域，后续部分则需要利用这些信息来解方程或不等式。
11. Exam Tips and Common Pitfalls | 考试技巧与常见误区
Many students lose marks because they omit the domain of a function, forget to verify that a function is one-to-one before finding its inverse, or use the wrong order of transformations. The following tips will help you avoid these errors.
许多学生因忽略函数定义域、求反函数前未验证函数是否一一对应，或变换顺序错误而失分。以下技巧可以帮助你避免这些错误。
- Always write the domain and range in interval notation where appropriate.
- When using a calculator to find intersections, state the equation being solved.
- For rational functions, check both vertical and horizontal asymptotes before sketching.
- Remember that \(\sqrt{x^2}=|x|\), not \(x\).
- When \(f\) is odd, \(f(0)=0\) if 0 is in the domain.
- 适当情况下，始终用区间记号写明定义域和值域。
- 使用计算器求交点时，写明所解的方程。
- 画有理函数图像前，先检查垂直和水平渐近线。
- 注意 \(\sqrt{x^2}=|x|\)，而不是 \(x\)。
- 若 \(f\) 是奇函数且0在定义域内，则 \(f(0)=0\)。
Practice by rewriting each function property from memory and applying it to a simple example. This active recall will strengthen your understanding and speed up your problem solving.
练习时，尝试凭记忆写出每个函数性质，并用简单例子验证。这种主动回忆法能加深理解并提高解题速度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导