Rational Functions and Zeros · 有理函数与零点
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rational function/ˈræʃənl ˈfʌŋkʃn/ | 有理函数 | yǒu lǐ hán shù |
| zeros/ˈzɪərəʊz/ | 零点 | líng diǎn |
| numerator/ˈnjuːməreɪtə/ | 分子 | fèn zǐ |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
| denominator/dɪˈnɒmɪneɪtə/ | 分母 | fēn mǔ |
When is a fraction zero?
- Think of $\frac{0}{5}$, $\frac{5}{5}$, and $\frac{5}{0}$: only the first is zero.
- A fraction is zero exactly when its top is zero — the bottom just cannot be zero too.
- Rational functions follow the same simple rule.
- So finding their zeros means looking only at the numerator.
分数什么时候等于零?
- 想想 $\frac{0}{5}$、$\frac{5}{5}$ 和 $\frac{5}{0}$:只有第一个等于零。
- 一个分数等于零,当且仅当它的分子为零——而且分母不能也为零。
- 有理函数遵循同样简单的规则。
- 所以求它们的零点,只需看分子。
Zeros come from the numerator
- A rational function 有理函数 is $\frac{\text{numerator}}{\text{denominator}}$.
- Its zeros 零点 are the inputs that make the whole thing equal $0$.
- That happens only where the numerator 分子 equals $0$.
- So set the top equal to zero and solve — those are your candidate x-intercepts.
零点来自分子
- 有理函数(rational function)是 分子 / 分母。
- 它的零点(zeros)是使整个式子等于 $0$ 的那些输入。
- 这只发生在分子(numerator)等于 $0$ 的地方。
- 所以令分子等于零并求解——那些就是你的候选 x 轴交点。
A fraction equals zero · 零 exactly when… · 一个分数等于零,当且仅当……
A fraction is zero only when its top is zero (and its bottom is not). $\frac{0}{5} = 0$, but $\frac{5}{0}$ is undefined. · 分数只有在分子为零(且分母不为零)时才等于零。$\frac{0}{5} = 0$,但 $\frac{5}{0}$ 无定义。
…but only if they are allowed
- A candidate zero counts only if the value is in the domain 定义域 of the function.
- If it also makes the denominator 分母 zero, the function is undefined there — no zero.
- So check each numerator zero against the bottom before you trust it.
……但前提是它们被允许
- 一个候选零点只有当它在函数的定义域(domain)内时才算数。
- 如果它同时也让分母(denominator)为零,函数在那里就无定义——不是零点。
- 所以在相信每个分子零点之前,先对照分母检查一下。
Where does it cross, and where does it break? · 它在哪里穿过,又在哪里断开?
y = a/(x − b) + c
The curve crosses the x-axis at its zero, but breaks at x = b where the denominator is zero. Those are two different places. · 曲线在它的零点处穿过 x 轴,却在 x = b(分母为零)处断开。这是两个不同的地方。
A numerator zero is a real zero of the function only if that value lies in the ____ of the function. · 分子的零点只有当它落在函数的____内时,才是函数真正的零点。
If a value makes both top and bottom zero, it is not in the domain, so it is a hole, not a zero. · 如果一个值让上下同时为零,它就不在定义域内,所以那是一个空心,而不是零点。
Excluded values are not zeros
- A value that makes only the denominator zero is excluded from the domain.
- There the function is undefined — it produces a vertical asymptote or a hole, not a zero.
- Never call an excluded value an x-intercept.
- Zero on top → a zero of the function; zero on the bottom → trouble, not a zero.
被排除的值不是零点
- 一个只让分母为零的值,被排除在定义域之外。
- 函数在那里无定义——它产生一条竖直渐近线或一个空心,而不是零点。
- 绝不要把被排除的值称为 x 轴交点。
- 分子为零 → 函数的零点;分母为零 → 麻烦,而不是零点。
A value that makes the denominator zero is a zero of the function. · 一个使分母为零的值,是函数的零点。
A denominator zero is excluded from the domain — the function is undefined there, not zero. · 使分母为零的值被排除在定义域之外——函数在那里无定义,而不是等于零。
Verify by evaluating
- To confirm a zero, substitute the value and check that the function really gives $0$.
- $f(x) = \frac{x-4}{x+1}$ at $x = 4$: $\frac{0}{5} = 0$. Yes — $x = 4$ is a zero.
- The same test rejects an excluded value, because you would be dividing by zero.
- Evaluating is the reliable final check.
用求值来验证
- 要确认一个零点,把值代入,检查函数是否真的给出 $0$。
- $f(x) = \frac{x-4}{x+1}$ 在 $x = 4$:$\frac{0}{5} = 0$。对——$x = 4$ 是零点。
- 同样的检验会否定一个被排除的值,因为那样你会除以零。
- 求值是可靠的最后一步检查。
How many zeros does $f(x) = \dfrac{(x-3)(x+2)}{x^2+1}$ have? · $f(x) = \dfrac{(x-3)(x+2)}{x^2+1}$ 有多少个零点?
The numerator is zero at $x = 3$ and $x = -2$; the denominator $x^2+1$ is never zero, so both are valid — $2$ zeros. · 分子在 $x = 3$ 和 $x = -2$ 处为零;分母 $x^2+1$ 永不为零,所以两个都成立——共 $2$ 个零点。
Select all · 所有 true statements about zeros of a rational function. · 选出关于有理函数零点的所有正确说法。
A denominator zero is excluded, not an x-intercept. The other three are correct. · 分母的零点是被排除的,不是 x 轴交点。其余三条正确。
Do not confuse a numerator zero (a real zero, an x-intercept) with a denominator zero (an excluded value). They are opposite: one is where the graph touches the axis, the other is where the graph breaks.
不要把分子的零点(真正的零点,x 轴交点)与分母的零点(被排除的值)搞混。它们正好相反:一个是图像接触轴的地方,另一个是图像断开的地方。
Find the zeros of $f(x) = \dfrac{(x-2)(x+3)}{x-2}$.
- Numerator zero candidates: $x = 2$ and $x = -3$.
- But $x = 2$ also makes the denominator zero, so it is excluded — a hole, not a zero.
- Only $x = -3$ is a real zero of the function.
求 $f(x) = \dfrac{(x-2)(x+3)}{x-2}$ 的零点。
- 分子零点候选:$x = 2$ 和 $x = -3$。
- 但 $x = 2$ 同时让分母为零,所以它被排除——是空心,不是零点。
- 只有 $x = -3$ 是函数真正的零点。
The zeros of a rational function are the zeros of its numerator that lie in the domain. A value that makes the denominator zero is excluded, not a zero. Always check numerator zeros against the bottom, and verify by evaluating.
有理函数的零点是它的分子在定义域内的零点。一个使分母为零的值是被排除的,不是零点。永远要对照分母检查分子零点,并用求值来验证。