这部分需要会什么？
- 理解输入、输出与函数值
- 正确代入整个代数式
- 按顺序计算复合函数并检查定义域
核心概念与公式
\(f(a)\) 表示把输入 \(a\) 代入函数 \(f\) 得到的输出，不表示 \(f\) 乘以 \(a\)。求 \(f(x+h)\) 时，原式中每一个 \(x\) 都要整体替换成 \((x+h)\)。
复合函数 \(f(g(x))\) 先计算内层 \(g(x)\)，再把其结果输入 \(f\)。合法输入既要属于 \(g\) 的定义域，又要让 \(g(x)\) 属于 \(f\) 的定义域。函数的同一输入只能对应一个输出。
做题时怎样使用
- 明确需要代入的是一个数，还是一个完整表达式
- 为替换进去的表达式加括号，处理平方、负号和分母
- 复合函数从内到外计算，并核对每层输入是否合法
容易混淆的地方
一般 \(f(x+h)\ne f(x)+h\)，且 \(f(g(x))\ne g(f(x))\)。求 \(f(a)=b\) 中的 \(a\) 是求输入，不能只计算已知输入的输出。
用一个例题串起方法
知识点教学例题已知 \(f(x)=2x^2-1\)，\(g(x)=x+3\)。求 \(f(g(1))\) 与 \(g(f(1))\)。
展开解题思路 ＋
\(g(1)=4\)，所以 \(f(g(1))=f(4)=31\)。而 \(f(1)=1\)，所以 \(g(f(1))=g(1)=4\)。复合顺序改变，结果也可能改变。
本知识点题库
先独立作答，再查看答案与解析。
函数记号・代入与复合进阶
The function h is defined by \( h\left(x\right)=5\left(9x-10\right) \). What is the value of \( h(10) \) ?
原库参考解析
函数记号・代入与复合进阶
The function \( A\left(b\right)=7b^2 \) gives the area of a right triangle, in square centimeters (cm²), if the length of its base is b cm and the length of its height is 14 times the length of its base. Which of the following is the best interpretation of \( A\left(18\right)=2,268 \)?
原库参考解析
函数记号・代入与复合基础
The function \( f\left(x\right)=-40x+340 \) gives the predicted height above the ground \( f\left(x\right) \), in feet, of a model airplane x minutes after it begins to descend. What is the predicted height above the ground, in feet, of the model airplane 3 minutes after it begins to descend?
原库参考解析
函数记号・代入与复合基础
The function \( f \) is defined by \( f\left(x\right)=2x^3+\frac{1}{3}x^2-x \). What is the value of \( f(1) \)?
原库参考解析
函数记号・代入与复合基础
Function \( f \) is defined as \( f\left(x\right)=\left(2x-5\right)^2 \), and function g is defined as \( g\left(x\right)=2\left(2x+7\right) \). Which of the following is equivalent to \( f\left(x\right)+g\left(x\right) \) ?
原库参考解析
函数记号・代入与复合挑战
\( f(x)=2x+3 \)
\( g(x)=7x-2 \)
\( h(x)=5x+6 \)
The functions \( f \), g , and h are defined as shown. If \( f(x)·g(x)−h(x)=ax^{2}+bx+c \) , where a, b , and c are constants, what is the value of b?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) is defined by \( f(x)=\frac{x+11}{5} \), and \( f(a)=-18 \), where a is a constant. What is the value of a?
原库参考解析
函数记号・代入与复合进阶
The function g is defined by \( g(x)=4x+2 \). What is the value of \( g(x) \) when \( x=2 \)?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by the equation \( f(x)=5x-6 \). What is the value of \( f(x) \) when \( x=10 \)?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=-3x+40 \). What is the value of \( f(x) \) when \( x=-7 \)?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) is defined by \( f(x)=6(x-2)(x^2-k) \), where k is a constant. In the xy-plane, the graph of \( y=f(x) \) passes through the point \( (-1,\ 80) \). What is the value of \( f(0) \)?
原库参考解析
函数记号・代入与复合进阶
The function \( f(w)=6w^2 \) gives the area of a rectangle, in square feet \( \left(ft^2\right) \) , if its width is w ft and its length is 6 times its width. Which of the following is the best interpretation of \( f(13)=1,014 \) ?
原库参考解析
