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
Which expression is equivalent to \( \frac{34π}{π^{2}−x^{2}}−\frac{34}{x+π} \) where \( x>π \)?
原库参考解析
分式方程与有理函数基础
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
分式方程与有理函数挑战
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, 27), and\( g(-27)=0 \) . What is the value of b?
原库参考解析
分式方程与有理函数挑战
\( \frac{1}{x-3}=\frac{x-1}{-x+1.75} \)
Which of the following is a solution to the given equation?
原库参考解析
分式方程与有理函数进阶
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
分式方程与有理函数进阶
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 \( v(x)=\frac{x^2+bx+c}{(x+5)(x−17)} \), 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
分式方程与有理函数挑战
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
分式方程与有理函数挑战
Which expression is equivalent to\( \frac{x^{2}−19x+70}{(x−5)(x+5)} \) , where \( x>5 \) ?
参考答案
分式方程与有理函数进阶
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)?
参考答案
分式方程与有理函数挑战
\( \frac{x+a}{x^2-a^2}=\frac{-2}{25x} \)
In the given equation, a is a constant. The solution to this equation is 6. What is the value of a?
In the given equation, a is a constant. The solution to this equation is 6. What is the value of a?
参考答案
