الدوال
دبلومة البكالوريا الدولية · IB Diploma · Mathematics: Applications and Interpretation · HL الموضوع 2 3:27 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
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النص الصوتي
These are selected topics, not the whole syllabus, and every example is fictional.
这些是精选主题，不是完整大纲，所有例子都是虚构的。
A function comes with its domain; in f g, g acts first; only a one-to-one function has an inverse.
函数要说明定义域；fg 中 g 先作用；只有一一对应的函数才有反函数。
A fitted line predicts within its data, and a residual is observed minus predicted.
拟合直线只在数据范围内预测；残差 = 观测值 − 预测值。
Exponential models change by the same factor each step, and logarithms undo them.
指数模型每一步按同一倍数变化，对数可以把它还原。
Root of x minus 2 exists only for x at least 2, and its outputs start at 0.
√(x − 2) 只在 x ≥ 2 时有定义，输出从 0 开始。
To invert, write y equals root of x minus 2 and solve for x: x is y squared plus 2.
求反函数：写 y = √(x − 2)，解出 x = y² + 2。
So f inverse is x squared plus 2, but only for x at least 0: the old range becomes the new domain.
所以 f⁻¹(x) = x² + 2，但只限 x ≥ 0：原来的值域变成新的定义域。
The two graphs are reflections in the line y equals x.
两个图像关于直线 y = x 对称。
Does x squared have an inverse?
x² 有反函数吗？
Draw a horizontal line at y equals 1: it meets the parabola twice, at 1 and minus 1.
画水平线 y = 1：它与抛物线交于两点，1 和 −1。
Two inputs share one output, so the function is many-to-one and cannot be undone.
两个输入对应同一个输出，所以函数是多对一的，无法还原。
Restrict the domain to x at least 0: now every horizontal line meets it once, and the inverse is root x.
把定义域限制为 x ≥ 0：每条水平线只交一次，反函数是 √x。
A line predicts bus time from distance: t hat equals 8 plus 2 d.
一条直线由距离预测公交时间：t̂ = 8 + 2d。
At 5 kilometres it predicts 18 minutes.
5 千米时预测 18 分钟。
The observed time was 21, so the residual, observed minus predicted, is 3 minutes: the vertical gap to the line.
实际是 21 分钟，所以残差（观测减预测）是 3 分钟：到直线的竖直距离。
The intercept 8 means minutes before moving; the slope 2 is minutes per kilometre.
截距 8 是出发前的分钟数；斜率 2 是每千米分钟数。
Far beyond the data, even a strong fit is unreliable.
远离数据时，即使拟合很好也不可靠。
A deposit of 1000 yuan earns 5 percent compound interest a year.
1000 元存款年复利 5%。
Each year multiplies by 1.05: 1050, then 1102.50, then 1157.63.
每年乘 1.05：1050，然后 1102.50，然后 1157.63。
Each year's interest is larger than the last, because it is 5 percent of a larger balance.
每年的利息都比前一年多，因为它是更大余额的 5%。
For a loan, separate the principal, the rate, the repayment and the period.
贷款要分清本金、利率、还款额和期数。
A medicine concentration follows y equals 80 e to the minus 0.2 t.
某药物浓度满足 y = 80e^(−0.2t)。
It loses the same fraction each hour, so it never goes negative.
它每小时失去相同的比例，所以永远不会变成负数。
When does it halve to 40?
何时减半到 40？
Divide: e to the minus 0.2 t is a half.
相除：e^(−0.2t) = ½。
Take logs: t is ln 2 over 0.2, about 3.47 hours.
取对数：t = ln2/0.2，约 3.47 小时。
After another 3.47 hours it halves again, to 20.
再过 3.47 小时又减半，到 20。
Take natural logs of y equals 80 e to the minus 0.2 t: ln y equals ln 80 minus 0.2 t.
对 y = 80e^(−0.2t) 取自然对数：ln y = ln80 − 0.2t。
Plot ln y against t and the curve becomes a straight line, gradient minus 0.2, intercept ln 80, about 4.38.
以 ln y 对 t 作图，曲线变成直线，斜率 −0.2，截距 ln80，约 4.38。
That is how data are tested for an exponential model.
这就是检验数据是否符合指数模型的方法。
But ln of a sum is not the sum of the logs.
但和的对数不等于对数的和。
Check yourself.
自我检测。
f g of 1 is f of 4, which is root 2.
fg(1) = f(4) = √2。
Predicted 18, observed 21: a residual of 3 minutes.
预测 18，观测 21：残差 3 分钟。
3 to the x equals 20 gives x as ln 20 over ln 3, about 2.73.
3ˣ = 20，x = ln20/ln3，约 2.73。