IGCSE数学 函数 复合函数 反函数 图像变换
CAIE 的 IGCSE 数学 0580 把函数放在代数与图像这条主线上,Core 阶段读线性与二次函数的图像,Extended 阶段加上函数记号、定义域与值域、复合函数、反函数。2025 年版考纲(适用于 2025 至 2027 年的考试)把定义域和值域明确写进 Extended 内容。函数题的分数很少丢在算不出来上,更多丢在顺序、范围和写法上。
Cambridge IGCSE Mathematics 0580 places functions on the algebra-and-graphs spine of the course. Core candidates read and sketch linear and quadratic graphs, while Extended candidates also handle function notation, domain and range, composite functions and inverses. The 2025 syllabus, examined from 2025 to 2027, states domain and range as Extended content in its own right. Marks are lost in the order of operations, the restrictions and the written form far more often than in the arithmetic.
函数就是把输入变成输出的规则
函数描述的是一条规则:每一个输入只对应一个输出。0580 用 f(x) = 3x - 5 这样的写法表示它,括号里是输入,等号右边是运算。求 f(4) 就是代入:3 乘 4 减 5 得 7;已知 f(a) = 16,问题就变成解 3a - 5 = 16,得 a = 7。很多学生以为 f(x) 里的 f 是一个数或一个变量,其实它只是函数的名字,换成 g 或 h,规则不变,图像也不变。常见的错误是把 f(3) 读成 f 乘 3,或者把 f(2x) 写成 2f(x),后者只在 f(x) = kx 这种没有常数项的形式下才成立。函数与一般关系的区别也常考:一个输入对应两个输出就不是函数,x² + y² = 25 的图像是一个圆,它写不成 y = f(x) 的形式。
A function is a rule that turns each input into exactly one output. In 0580 it usually appears as f(x) = 3x - 5, where the bracket holds the input and the right-hand side says what to do with it. Evaluating f(4) is plain substitution, whereas solving f(a) = 16 turns the question into an equation and gives a = 7. The letter f is a label, not a quantity, so renaming it g or h changes nothing about the rule or the graph. Beware of rewriting f(2x) as 2f(x): that shortcut only works for rules of the form kx with no constant term.
定义域与值域:2025 年新增的考查点
定义域是所有允许输入的集合,值域是所有可能输出的集合。Cambridge International 在 2025 年版 0580 考纲里把 domain and range 加进 Extended 内容,官网的教师说明 Getting ready to teach 把这一条列在新增知识点里,所以它从 2025 年起成为正式考点。求定义域主要避开三类禁区:分母为零、偶次根号内为负,以及题目自己给出的限制,例如 f(x) = 1/(x - 3) 的定义域是 x 不等于 3,g(x) = √(x + 2) 要求 x 大于或等于 -2。值域通常靠图像或配方来读,配方之后最小值一眼可见,反比例函数的值域则要说明 y 不等于 0。
The domain lists every input a function accepts, and the range lists every output it can produce. Cambridge International added domain and range to the Extended content when it revised 0580 for first assessment in 2025, and the official Getting ready to teach factsheet on its website lists the topic among the additions. Two bans do most of the work, division by zero and the square root of a negative number, plus any restriction the question itself imposes. For y = a/x the range excludes zero. Reading the range off a sketch or a completed square is faster and safer than listing outputs one by one.
复合函数 gf(x):先算里层再算外层
复合函数表示把一个函数的输出当成另一个函数的输入,写作 gf(x) = g(f(x)),所以顺序永远是从右往左算。设 f(x) = 3x - 5,g(x) = x² + 1,那么 gf(2) 先算 f(2) = 1,再算 g(1) = 2;如果题目要的是 gf(x) 的表达式,就把 f(x) 整块代进 g 的 x 位置,得 (3x - 5)² + 1。很多学生以为 gf 与 fg 结果一样,把数字代进去就会发现两者不同,只有极少数的特殊函数才碰巧相等。常见的错误是漏掉括号,把 (3x - 5)² 展开成 3x² - 5,平方作用在整个括号上,不是只作用在 x 上。复合之后的定义域还要兼顾里层的限制:若 f(x) = 1/x 与 g(x) = x - 1 复合,x 不能等于 1,这类边界在 Extended 题目里会单独设问。
A composite function feeds one rule into another, so gf(x) means g(f(x)) and you work from the inside outwards. Taking f(x) = 3x - 5 and g(x) = x² + 1, the value gf(2) comes from f(2) = 1 followed by g(1) = 2, while the algebraic form needs the whole expression 3x - 5 put wherever g contains an x. Order matters here: swap the two functions and the answer almost always changes. Missing brackets cause most of the damage, because (3x - 5)² squares the entire bracket rather than x alone.
反函数 f^-1(x):换主元再解方程
反函数把输出送回输入,记作 f^-1(x)。0580 Extended 要求会求它的表达式,方法只有一条主线:先写 y = f(x),交换 x 与 y,解出 y,最后把 y 换成 f^-1(x)。以 f(x) = (x - 3)/2 为例,交换后得 x = (y - 3)/2,两边乘 2 再移项,f^-1(x) = 2x + 3。检验只要十秒:f(7) = 2,而 f^-1(2) = 7,回到起点说明式子对了。学生最常把 f^-1(x) 当成 1/f(x),这是记号上的陷阱,负一次方在这里表示反过来,不表示倒数,写成倒数在阅卷时直接判错。
An inverse function undoes the original mapping and is written f^-1(x). The routine never changes: write y = f(x), swap x and y, rearrange for y, then relabel. For f(x) = (x - 3)/2 the swap gives x = (y - 3)/2, so doubling both sides and tidying up yields f^-1(x) = 2x + 3. Checking takes ten seconds, since f(7) = 2 combined with f^-1(2) = 7 proves the pair really are inverses. Reading f^-1(x) as the reciprocal 1/f(x) is a notation trap that examiners see every series.
配方与顶点:二次函数图像的读法
二次函数 y = ax² + bx + c 的图像是抛物线,配方能把顶点直接读出来。把 y = x² - 6x + 5 写成 y = (x - 3)² - 4,顶点就是 (3, -4),对称轴是 x = 3,与 x 轴的两个交点解 (x - 3)² - 4 = 0 得 x = 1 和 x = 5。系数不为 1 时先提出 a:y = 2x² + 8x + 1 变成 2(x + 2)² - 7,顶点是 (-2, -7)。很多学生以为顶点横坐标就是配方括号里的常数,忘记要变号,于是把 (3, -4) 写成 (-3, -4)。0580 Core 的 C2.11 只要求对称性和根,顶点的完整读法属于 Extended 的常见考法。
A quadratic graph is a parabola, and completing the square exposes its vertex in one line. Rewriting y = x² - 6x + 5 as y = (x - 3)² - 4 gives the vertex (3, -4), the axis of symmetry x = 3 and the roots x = 1 and x = 5. When the leading coefficient is not 1, factor it out first: y = 2x² + 8x + 1 becomes 2(x + 2)² - 7 with vertex (-2, -7). The bracket constant flips sign when it appears in the coordinates, and that missing flip costs marks in every session. Cambridge's Core outcomes ask for symmetry and roots only, while turning points sit on the Extended path.
常数在图像上的作用:平移、拉伸与反射
已知 y = f(x) 的图像,加在括号外的常数把整条曲线上下移动,加在括号里的常数把曲线左右移动,而且方向与直觉相反。y = f(x) + k 向上平移 k 个单位,y = f(x + k) 向左平移 k 个单位。括号外的系数 a 把纵坐标拉伸 a 倍,a 为负数时图像上下翻转;把 x 换成 -x 则是沿 y 轴左右翻转。做题时先标出原图的关键点,顶点、交点、渐近线各要一个,再逐点做变换,比凭感觉重画快。学生最常把 y = f(x + 3) 画成向右平移三格,实际上它的顶点比原图靠左三格。稳妥的做法是把变换拆成两步:先处理括号内的平移,再处理括号外的拉伸或反射,顺序写反会让顶点坐标整个算错。
With the graph of y = f(x) given, a constant added outside the bracket shifts the curve vertically, while one added inside shifts it horizontally, and the horizontal direction runs opposite to intuition. Multiplying the output by a stretches the graph vertically, with a negative factor flipping it, whereas replacing x with -x mirrors the picture in the y-axis. Pick three or four landmarks, such as the vertex, the intercepts and any asymptote, and move those points instead of redrawing the whole curve. Writing f(x + 3) shifts left by three, not right.
反比例与指数图像:a/x 与 a^x 的形状和渐近线
函数图像题里有两类曲线反复出现。y = a/x(x 不为 0)的图像叫双曲线,当 a 大于 0 时两支分别落在第一和第三象限,坐标轴是它的渐近线,曲线无限接近但永不相交。y = a^x(a 为正整数)是指数曲线,必过点 (0, 1),当 a 大于 1 时从左到右迅速上升,而 x 越负曲线越贴近 x 轴。考纲 E2.10 要求先列取值表再描点作图,列一张 -2 到 3 的表比直接手绘稳妥,形状与渐近线的位置都能对上。常见的错误是把 y = a/x 画成穿过原点的直线,或把指数曲线画成抛物线。
Two families of curves dominate graph questions. The reciprocal graph y = a/x splits into two branches in opposite quadrants, with the axes acting as asymptotes that the curve approaches but never reaches. The exponential graph y = a^x always passes through (0, 1), climbs steeply to the right for a greater than 1, and hugs the x-axis on the left. Cambridge's outcome E2.10 asks for a table of values before drawing, and that minute of arithmetic removes most shape errors. Straight lines through the origin and parabolic exponentials are the two mistakes that appear most often in weak scripts.
用函数建立模型:文字题里的输入与输出
文字题把函数藏在情境里,比如出租车起步价加每公里单价、水箱里的水量随时间变化、正方形边长增加后面积的增长。做法是先找出随什么变化,把它当作输入 x,再把题目最终要的量当作输出,写出表达式。一家工厂固定成本 300 元、每件成本 12 元,总成本函数就是 C(n) = 12n + 300;问成本不超过 900 元最多能做多少件,解 12n + 300 ≤ 900 得 n ≤ 50。实际情境还会限制定义域:件数是整数,时间不会是负数,把这类限制写出来往往本身就能得分,Extended 的定义域题也常从这里追问。
Word problems hide a rule inside a story, and the first step is to decide what is changing and what is being asked for. A workshop with fixed costs of 300 and 12 per item is described by C(n) = 12n + 300, so keeping total cost within 900 becomes the inequality 12n + 300 ≤ 900 and gives n ≤ 50. Real contexts also trim the domain, since counts of items are whole numbers and elapsed time cannot be negative. Stating those restrictions explicitly can earn a mark on its own, and Extended domain questions often press exactly there.
非计算器卷的函数题怎么练
从 2025 年起 0580 每个层级都有一份非计算器卷:Core 的 Paper 1 与 Extended 的 Paper 2 都不允许使用计算器,两份卷各占该层级总分的一半,时长分别是 1 小时 30 分钟与 2 小时,原来的计算器卷 Paper 3 与 Paper 4 各 80 分与 100 分。这一变化让函数题里的口算与分数运算变成硬功夫。练的方法是代入和配方都手算一遍,算 f(-3) 时先算 3 乘 -3 再减 5 得 -14,配方遇到分数系数就通分,不要写成小数。所有卷子的第二页提供公式表,考纲也说明并非每一个公式都会给出,记住哪些要背能省下考场上的犹豫。
Since 2025 each tier of 0580 includes a non-calculator paper: Paper 1 for Core and Paper 2 for Extended, worth half of the tier total, with durations of 1 hour 30 minutes and 2 hours. Both the calculator papers, Paper 3 and Paper 4, are set at 80 and 100 marks. Function questions therefore reward fluent mental and written arithmetic, so practise substitution and completing the square by hand and keep answers as exact fractions instead of rounded decimals. A list of formulas sits on page 2 of every paper, but the syllabus is explicit that not every result is supplied, so know which ones must stay in your head.
阅卷里最常见的丢分点
函数题的分数常常丢在写法上。阅卷时反复看到三类问题:求反函数只写到交换的那一步,没有整理成 f^-1(x) = 的形式;求复合函数漏掉括号,把 (x + 2)² 展开成 x² + 4;定义域题只写一个限制,忘了分母不能为零。命令词也要读准,show that 要求写出完整过程并给出结论句,find 给出答案即可,两者的过程分与答案分分配不同,过程写得潦草会被扣分。还有一种失分发生在画图上,曲线形状对了却没有标注顶点、交点或渐近线,满分同样拿不到。
Marks in function questions disappear in the written form rather than in the thinking. Scripts regularly stop halfway through an inverse, leaving the swap without a tidy f^-1(x) expression, or expand (x + 2)² as x² + 4. Domain answers frequently list one restriction and forget the other. Command words carry instructions too: show that wants full working and a closing statement, while find accepts a correct answer with less scaffolding. On graph questions, annotate the vertex, the intercepts and any asymptote, because an unlabelled curve seldom earns full credit.
六周复习排期:从图像到复合函数的推进顺序
六周时间够把函数这条线走完。前两周打基础,弄清代入、定义域与值域、配方,每天做三到五道课本例题,每一步都写完整。第三到四周转入图像,用同一族函数取五个不同参数各画一次,对照顶点、交点与渐近线检查形状和平移方向。第五周专攻复合函数与反函数,把 gf 与 fg 的结果并列写出,再用具体数字检验顺序。最后一周做近年真题里的函数题并限时完成,错题按计算错、范围错、写法错三类分开整理,错题本每两周回看一次,比重刷新题更有用。
Six weeks is enough to work through functions in a planned order. Spend the first fortnight on substitution, domain, range and completing the square, writing every step in full. Then move to graphs, drawing one family five times with different parameters and checking vertices, intercepts and asymptotes against each sketch. Composite and inverse functions deserve a week of their own, with gf and fg written side by side and tested with numbers. Finish on timed past-paper questions, sorting your errors into arithmetic, range and presentation, and revisit that log every fortnight.
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