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
If \( |\ 12-2x\ |=48 \), which of the following is a value of \( 6−x \)?
原库参考解析
分式・有理函数与绝对值进阶
\( \frac{|\ 3x-54\ |+3}{6}=9 \)
What is the sum of the solutions to the given equation?
参考答案
分式・有理函数与绝对值进阶
The function \( f \) is defined by \( f(x)=\frac{ax+b}{x+c}+k \) where a, b, c and k are constants. The graph of \( y=f(x) \) is shown. For how many values of x does \( f(x)=0 \)?
参考答案
分式・有理函数与绝对值进阶
\( \frac{|4x-48|+4}{6}=8 \)
What is the sum of the solutions to the given equation?
参考答案
分式・有理函数与绝对值进阶
The expression \( \frac{x+7}{x^2+2x-35}-\frac{16-x^2}{x+4} \) is equivalent to \( \frac{x^2+bx+c}{x-5} \), where b and c are constants and \( x>5 \). What is the value of bc?
参考答案
分式・有理函数与绝对值进阶
\( ∣2x−3∣=−8 \)
How many solutions does the given equation have?
参考答案
分式・有理函数与绝对值挑战
The function v is defined by the equation \( v(x)=\frac{x^2+bx+c}{(x+5)(x−17)} \), where b and c are constants. In the xy-plane, the graph of \( y=v(x) \) passes through the points (0, \( \frac{77}{85} \)) and (11, 0). If \( v(q)=0 \), which of the following could be the value of q?
参考答案
分式・有理函数与绝对值挑战
\( |x−9|=14 \)
What is the sum of the solutions to the given equation?
What is the sum of the solutions to the given equation?
参考答案
分式・有理函数与绝对值挑战
The table shows three values of x and their corresponding values of \( g(x) \), where \( g(x)=\frac{f(x)}{x+6} \) and \( f \) is a linear function. What is the y-coordinate of the y-intercept of the graph of \( y=f(x) \) in the xy-plane?
参考答案
分式・有理函数与绝对值挑战
Which expression is equivalent to\( \frac{x^{2}−19x+70}{(x−5)(x+5)} \) , where \( x>5 \) ?
参考答案
分式・有理函数与绝对值进阶
The function h is defined by \( h(x)=5|x| \). What is the value of h(-4)?
参考答案
分式・有理函数与绝对值进阶
\( 2|5x−5|=11 \)
What is the sum of the solutions to the given equation?
What is the sum of the solutions to the given equation?
参考答案
