这部分需要会什么？
- 先确定分式的定义域，再化简或解方程
- 区分约分后的空点与竖直渐近线
- 将绝对值理解为距离并分类求解
核心概念与公式
分式的原分母不能为零。约分只能消去公共因式，不能单独消去加减项；约分后仍要保留原式的限制。若 \((x-r)\) 被完全约去且剩余分母在 \(r\) 处非零，原函数在 \(x=r\) 处有空点。
\(|x-a|\) 表示 \(x\) 与 \(a\) 的距离。\(|x-a|=d\) 在 \(d>0\) 时有 \(x=a\pm d\)，\(d=0\) 时只有 \(x=a\)，\(d<0\) 时无解。\(|x-a|\le d\)（\(d\ge0\)）表示 \(a-d\le x\le a+d\)。
做题时怎样使用
- 分式先写出所有分母不为零的限制，再通分、约分或交叉相乘
- 绝对值先判断右侧正负，再写对应的两个分支或距离区间
- 最后检查候选解是否属于原定义域和对应分支
容易混淆的地方
把 \((x^2-1)/(x-1)\) 约成 \(x+1\) 后仍需 \(x\ne1\)。绝对值方程有时只有一个解或无解，不能一律说有两个。
用一个例题串起方法
知识点教学例题解 \(|2x-3|=5\)。
展开解题思路 ＋
分成 \(2x-3=5\) 或 \(2x-3=-5\)。因此 \(x=4\) 或 \(x=-1\)。两者代回绝对值均为 5。
本知识点题库
先独立作答，再查看答案与解析。
分式・有理函数与绝对值挑战
f (x) = | 71 - 2x |
The function f is defined by the given equation. For which of the following values of k does f(k) = 3k ?
原库参考解析
分式・有理函数与绝对值挑战
\( \frac{\left(x-3\right)\left(x-6\right)}{x-2}=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式・有理函数与绝对值进阶
\( \frac{27\left(k+6\right)}{3\left(k+6\right)} \)
The given expression is equivalent to c, where c is a constant and \( k>−6 \) . What is the value of c?
原库参考解析
分式・有理函数与绝对值挑战
\( \frac{1}{5xy}+xyz=\frac{1}{4yz} \)
In the given equation, x, y, and z are positive numbers. Which expression is equivalent to y?
原库参考解析
分式・有理函数与绝对值挑战
The equation \( 2|x-5|=k, \)where k is a constant, has exactly one solution. Which of the following could be the value of\( \frac{k}{2} \) ?
原库参考解析
分式・有理函数与绝对值进阶
The function g is defined as\( g\left(x\right)=\frac{2x-8}{\left(x+13\right)\left(x-6\right)} \) . If \( g(a+7)=0, \) where a is a constant, what is the value of a?
原库参考解析
分式・有理函数与绝对值进阶
\( |x-9|=-|x-9|+10 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式・有理函数与绝对值挑战
\( \frac{1}{R}=\frac{1}{p}+\frac{1}{q}+\frac{1}{s}+\frac{1}{32} \)
A parallel electric circuit contains four resistors with resistances of p ohms, q ohms, s ohms, and 32 ohms. The given equation relates the resistance R, in ohms, of the circuit, to the resistances of the four individual resistors it contains. Which equation correctly expresses R in terms of p, q, and s?
原库参考解析
分式・有理函数与绝对值挑战
The function q is defined by \( q\left(x\right)=a\left|x-9\right|^2-87\left|x-9\right|+b \) , where a and b are constants, and a > b> 87. If q(439) = h and q(-421) = k , where h and k are constants, what is the value of\( 9\left(-439\right)^{h-k}+421\left(9\right)^{k-h} \) ?
原库参考解析
分式・有理函数与绝对值进阶
\( \left|x-6\right|=15 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式・有理函数与绝对值进阶
\( \frac{1}{x}+\frac{1}{x-9}=\frac{10}{x^2-9x} \)
What are all the solutions to the given equation?
原库参考解析
分式・有理函数与绝对值进阶
\( |\ x-8\ |\ =11 \)
What is the sum of the solutions to the given equation?
原库参考解析
