国際バカロレア · IB Diploma · Mathematics: Analysis and Approaches · HL トピック 2 2:22 英語ナレーション・英語＋中文字幕 burning-in
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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。
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 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。
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。