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
- 解完分式方程检验候选解是否使分母为零
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；若 \((x-r)\) 约不掉，\(x=r\) 处是竖直渐近线。
解分式方程常用通分或交叉相乘化为多项式方程，但由此得到的候选解必须代回原分母检查，任何使分母为零的解都要舍去。
做题时怎样使用
- 先写出所有分母不为零的限制，再通分、约分或交叉相乘
- 化成多项式方程后求出候选解
- 把候选解代回原分母检查，舍去使分母为零的增根
容易混淆的地方
把 \((x^2-1)/(x-1)\) 约成 \(x+1\) 后仍需 \(x\ne1\)。交叉相乘可能引入使原分母为零的增根，必须检验后舍去。
用一个例题串起方法
知识点教学例题解 \(\frac{1}{x}+\frac{1}{x-2}=\frac{2}{x^2-2x}\)。
展开解题思路 ＋
因为 \(x^2-2x=x(x-2)\)，定义域要求 \(x\ne0\) 且 \(x\ne2\)。两边乘 \(x(x-2)\) 得 \((x-2)+x=2\)，即 \(2x-2=2\)，\(x=2\)。但 \(x=2\) 使原分母为零，必须舍去，所以原方程无解。
本知识点题库
先独立作答，再查看答案与解析。
分式方程与有理函数挑战
The table shows three values of \( x \) and their corresponding values of \( f(x) \), where \( f\left(x\right)=\frac{kx+63}{x+3} \) and \( k \) is a constant. What is the value of \( k \)?
原库参考解析
分式方程与有理函数进阶
\( \frac{\left(x-7\right)\left(x-8\right)}{x-3}=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式方程与有理函数进阶
The function w is defined by \( w\left(r\right)=\frac{1}{r-8}-\frac{r-5}{-r+4.25} \). What is the greatest possible value of r such that \( w\left(r\right)=0 \) ?
原库参考解析
分式方程与有理函数进阶
\( \frac{\left(x-6\right)\left(x-8\right)}{x-5}=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
分式方程与有理函数进阶
The function w is defined by \( w(r)=\frac{1}{r-5}-\frac{r-4}{-r+4.25} \). What is the greatest possible value of r such that \( w(r)=0 \)?
原库参考解析
分式方程与有理函数进阶
The function \( f \) is defined by \( f(x)=\frac{1}{8x} \). What is the value of \( f(x) \) when \( x=9 \)?
原库参考解析
分式方程与有理函数进阶
The function j is defined by \( j(x)=\frac{10x+5}{3p} \), and the function k is defined by \( k(x)=\frac{j(x)·p}{x+1} \), where p is a positive constant. What is the value of \( k(9) \)?
原库参考解析
分式方程与有理函数挑战
The function v is defined by the equation \( v(x)=\frac{x^2+bx+c}{(x+5)(x-19)} \), where b and c are constants. In the xy-plane, the graph of \( y=v(x) \) passes through the points \( \left(0,\ \frac{66}{95}\right) \) and \( (11,\ 0) \). If \( v(q)=0 \), which of the following could be the value of q?
原库参考解析
分式方程与有理函数进阶
For a rational function \( f \), the table shows three values of x and their corresponding values of \( f(x) \). Which of the following could be the graph of \( y=f(x) \)?
原库参考解析
分式方程与有理函数进阶
The function \( f \) is defined by \( f(x)=\frac{2}{11-x} \). What is the value of \( f(x) \) when \( x=9 \)?
原库参考解析
分式方程与有理函数挑战
Which expression is equivalent to \( \frac{7x(x-9)-2(x-9)}{3x-27} \), where \( x>9 \)?
原库参考解析
分式方程与有理函数挑战
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
