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
What is the x-intercept of the graph of \( y=\frac{21-x}{ax+b} \) in the xy-plane, where a and b are positive constants?
参考答案
分式方程与有理函数基础
What is the x-intercept of the graph of \( y=\frac{21-x}{ax+b} \) in the xy-plane, where a and b are positive constants?
参考答案
分式方程与有理函数挑战
What is the x-intercept of the graph of \( y=\frac{28-x}{ax+b} \) in the xy-plane, where a and b are positive constants?
参考答案
分式方程与有理函数挑战
\( \frac{z+4}{5r^2}-\frac{5}{z+4}=0 \)
In the given equation, r is a positive constant. A solution to this equation is \( −84 \). What is the value of r?
In the given equation, r is a positive constant. A solution to this equation is \( −84 \). What is the value of r?
参考答案
分式方程与有理函数进阶
In the xy-plane, the graph of function \( f \), where \( y=f\left(x\right) \), has exactly \( 8 \) x-intercepts. One of these x-intercepts is \( \left(17,\ 0\right) \). The rational function g is defined by \( g\left(x\right)=\frac{f\left(x\right)}{x-17} \). In the xy-plane, how many x-intercepts does the graph of \( y=g\left(x\right) \) have?
原库参考解析
分式方程与有理函数挑战
The function \( j \) is defined by \( j(x)=\frac{4x+2}{9p} \), 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 \( k(3) \)?
原库参考解析
分式方程与有理函数进阶
If \( \frac{48}{x}=4x \), what is the value of \( 4x^2 \)?
原库参考解析
分式方程与有理函数挑战
\( g(x)=\frac{x^2-x-a}{x^2-x-b} \)
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, 43), and \( g(-43)=0 \) . What is the value of b?
原库参考解析
分式方程与有理函数挑战
Which expression is equivalent to \( \frac{3x(x-8)-9(x-8)}{2x-16} \), where \( x>8 \)?
原库参考解析
分式方程与有理函数挑战
Which expression is equivalent to \( \frac{y+12}{x-8}+\frac{y\left(x-8\right)}{x^2y-8xy} \) ?
原库参考解析
分式方程与有理函数挑战
\( g(x)=\frac{x^2-x-a}{x^3-x-b} \)
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, 22), and \( g(-22)=0 \). What is the value of b?
原库参考解析
分式方程与有理函数进阶
Which expression is equivalent to \( \frac{42a}{k}+42ak \) , where \( k \)>0?
原库参考解析
