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
Which expression is equivalent to \( \frac{7x(x-9)-2(x-9)}{3x-27} \), where \( x>9 \)?
原库参考解析
分式・有理函数与绝对值挑战
The equation\( 5|\ x-6\ |=k \) , where k is a constant, has exactly one solution. Which of the following could be the value of \( \frac{k}{5} \)?
原库参考解析
分式・有理函数与绝对值挑战
The table shows three values of x and their corresponding values of \( f(x) \), where \( f(x)=\frac{kx+45}{x+3} \) and k is a constant. What is the value of k?
原库参考解析
分式・有理函数与绝对值挑战
Which expression is equivalent to \( \frac{34π}{π^{2}−x^{2}}−\frac{34}{x+π} \) where \( x>π \)?
原库参考解析
分式・有理函数与绝对值基础
\( 5|\ x\ |=35 \)
What is the positive solution to the given equation?
原库参考解析
分式・有理函数与绝对值基础
What is the x-intercept of the graph of \( y=\frac{45-x}{ax+b} \) in the xy-plane, where a and b are positive constants?
原库参考解析
分式・有理函数与绝对值挑战
What is the x-intercept of the graph of \( y=\frac{45-x}{ax+b} \) in the xy-plane, where a and b are positive constants?
原库参考解析
分式・有理函数与绝对值进阶
The graph of a system of an absolute value function and a linear function is shown. What is the solution \( (x,y) \) to this system of two equations?
原库参考解析
分式・有理函数与绝对值挑战
\( g(x)=\frac{x^2-x-a}{x^3-x-b} \)
The function g is defined by the given equation, where a and b are constants. In the xy-plane, the graph of \( y=g(x) \) passes through the point (0, 27), and\( g(-27)=0 \) . What is the value of b?
原库参考解析
分式・有理函数与绝对值挑战
The equation \( 5|x-9|=k \) , where k is a constant, has exactly one solution. Which of the following could be the value of k?
原库参考解析
分式・有理函数与绝对值挑战
\( \frac{1}{x-3}=\frac{x-1}{-x+1.75} \)
Which of the following is a solution to the given equation?
原库参考解析
分式・有理函数与绝对值进阶
\( 6|x−6|=24 \)
What is the sum of the solutions to the given equation?
原库参考解析
