Rational Functions and Holes · 有理函数与空心
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| hole/həʊl/ | 空心 | kōng xīn |
| factor/ˈfæktə/ | 因式 | yīn shì |
| numerator/ˈnjuːməreɪtə/ | 分子 | fèn zǐ |
| denominator/dɪˈnɒmɪneɪtə/ | 分母 | fēn mǔ |
| removable discontinuity/rɪˈmuːvəbl dɪskɒntɪˈnjuːɪti/ | 可去间断点 | kě qù jiàn duàn diǎn |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
A single missing point
- Most rational graphs break dramatically at vertical asymptotes.
- But sometimes the graph is a perfectly ordinary line or curve — with just one point punched out.
- That tiny gap is called a hole, and it is easy to miss.
- It appears for a very specific algebraic reason.
一个缺失的点
- 大多数有理函数图像在竖直渐近线处剧烈地断开。
- 但有时图像是一条完全普通的直线或曲线——只是一个点被挖掉了。
- 那个小小的缺口叫做空心,很容易被忽略。
- 它的出现有一个非常明确的代数原因。
Where holes come from
- A hole 空心 appears when a factor 因式 cancels from both the numerator and the denominator.
- Take $\frac{(x-1)(x+2)}{(x-1)}$: the $(x-1)$ cancels, leaving $x+2$.
- The simplified graph is the smooth line $y = x + 2$.
- But at $x = 1$ the original fraction was $\frac{0}{0}$ — undefined — so a point is missing.
空心从哪来
- 当一个因式(factor)在分子和分母中同时被约掉时,就出现一个空心(hole)。
- 看 $\frac{(x-1)(x+2)}{(x-1)}$:$(x-1)$ 被约掉,剩下 $x+2$。
- 化简后的图像是光滑的直线 $y = x + 2$。
- 但在 $x = 1$ 处,原来的分数是 $\frac{0}{0}$——无定义——所以缺了一个点。
A hole in a rational graph appears where… · 有理函数图像中的空心出现在……
A common factor top and bottom cancels, so the simplified graph is smooth — but the original value is still excluded, leaving a hole. · 上下的公因式被约掉,所以化简后的图像是光滑的——但原来的那个值仍被排除,留下一个空心。
Hole versus vertical asymptote
- Both come from a denominator 分母 zero, but they behave oppositely.
- Cancels with the numerator 分子 → a removable discontinuity 可去间断点: a hole.
- Does not cancel → a vertical asymptote where the graph shoots off.
- So always factor and simplify first, then see which zeros cancel.
空心与竖直渐近线
- 两者都来自分母(denominator)的零点,但行为相反。
- 与分子(numerator)约掉 → 一个可去间断点(removable discontinuity):空心。
- 没有约掉 → 一条竖直渐近线,图像在那里飞出去。
- 所以永远先因式分解并化简,再看哪些零点被约掉。
The simplified function is just a line — with one point removed · 化简后的函数只是一条直线——但少了一个点
y = ax + b
This is the simplified form of (x²−1)/(x−1). The graph is the line y = x + 1, but the real function has a hole at x = 1 where the original was undefined. · 这是 (x²−1)/(x−1) 的化简形式。图像是直线 y = x + 1,但真正的函数在 x = 1 处有一个空心,因为原式在那里无定义。
What distinguishes a hole from a vertical asymptote? · 空心与竖直渐近线的区别是什么?
Cancelled factor → removable hole; surviving denominator zero → vertical asymptote where values blow up. · 被约掉的因式 → 可去的空心;残留的分母零点 → 竖直渐近线,值在那里爆炸。
To find a hole's coordinates, cancel the common factor, then evaluate the ____ function at the excluded value. · 要找空心的坐标,先约掉公因式,再把被排除的值代入____后的函数。
The simplified function gives the y-value the graph would have there — that is the hole's height. · 化简后的函数给出图像在那里本应有的 y 值——那就是空心的高度。
Finding the hole's coordinates
- Cancel the common factor to get the simplified function.
- Substitute the excluded x-value into that simplified function to get the y-value.
- For $\frac{(x-1)(x+2)}{x-1}$ at $x = 1$: simplified is $x+2$, giving $y = 3$.
- So the hole sits at $(1, 3)$ — the point the graph "should" have passed through.
求空心的坐标
- 约掉公因式,得到化简后的函数。
- 把被排除的 x 值代入那个化简后的函数,得到 y 值。
- 对 $\frac{(x-1)(x+2)}{x-1}$ 在 $x = 1$:化简后是 $x+2$,给出 $y = 3$。
- 所以空心位于 $(1, 3)$——图像"本应"经过的那个点。
Even after the factor cancels, the hole's x-value is still excluded from the domain. · 即使因式被约掉,空心的 x 值仍然被排除在定义域之外。
The original function was never defined there, so cancelling in algebra does not add the point back — the domain still has a gap. · 原函数在那里从未有定义,所以代数上的约分并不会把这个点加回来——定义域仍有一个缺口。
Select all · 所有 true statements about a hole. · 选出关于空心的所有正确说法。
A hole does not · 不 send the function to infinity — that is an asymptote. The other three are true. · 空心不会让函数冲向无穷——那是渐近线。其余三条正确。
Still excluded from the domain
- Cancelling is algebra; it does not restore the missing input.
- The original function was undefined at that x, so it stays out of the domain 定义域.
- The simplified formula agrees with the original everywhere except that one point.
- That is exactly what "removable" means: you can patch the single gap, but it was still there.
仍被排除在定义域之外
- 约分是代数操作;它并不能把缺失的输入恢复回来。
- 原函数在那个 x 处无定义,所以它仍在定义域(domain)之外。
- 化简后的式子与原式在除那一点外处处相等。
- 这正是"可去"的意思:你可以补上这个缺口,但它原本确实存在。
Simplifying $\frac{(x-1)(x+2)}{x-1}$ to $x+2$ is only valid for $x \neq 1$. Never drop the restriction — the two expressions are equal everywhere except at the hole, and forgetting that loses a point from the domain.
把 $\frac{(x-1)(x+2)}{x-1}$ 化简为 $x+2$ 只在 $x \neq 1$ 时成立。永远不要丢掉这个限制——两个式子在除空心外处处相等,忘了这一点就会从定义域里丢掉一个点。
Locate any hole of $f(x) = \dfrac{x^2 - 4}{x - 2}$.
- Factor the top: $\frac{(x-2)(x+2)}{x-2}$, and cancel $(x-2)$ → simplified $x + 2$.
- Excluded value $x = 2$; simplified value there is $2 + 2 = 4$.
- So there is a hole at $(2, 4)$, and no vertical asymptote.
找出 $f(x) = \dfrac{x^2 - 4}{x - 2}$ 的空心。
- 把分子因式分解:$\frac{(x-2)(x+2)}{x-2}$,约掉 $(x-2)$ → 化简为 $x + 2$。
- 被排除的值是 $x = 2$;化简后在那里的值是 $2 + 2 = 4$。
- 所以在 $(2, 4)$ 处有一个空心,而没有竖直渐近线。
A hole is a removable discontinuity: a factor that cancels from numerator and denominator. Find it by simplifying and evaluating at the excluded x-value. The x stays out of the domain — cancelling patches the graph but never adds the point back.
空心是一个可去间断点:一个在分子和分母中都被约掉的因式。通过化简并在被排除的 x 值处求值来找到它。那个 x 仍在定义域之外——约分补上了图像,却从不把点加回来。