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・代入与复合基础
The function f is defined by the equation f(x)=100x +2. What is the value of f(x) when x =9?
原库参考解析
函数记号・代入与复合挑战
The function h is defined by h(x)= x +200. What is the value of h(50)?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f\left(x\right)=80-6x \) . What is the value of \( f\left(7\right) \) ?
原库参考解析
函数记号・代入与复合挑战
The function \( f\left(x\right)=55.20-0.16x \) gives the estimated surface water temperature \( f\left(x\right) \), in degrees Celsius, of a body of water on the xth day of the year, where \( 220\le x\le360 \) . Based on the model, what is the estimated surface water temperature, in degrees Celsius, of this body of water on the 326th day of the year?
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) =5x +8. For what value of x does f(x) =58?
原库参考解析
函数记号・代入与复合基础
The function \( f \) is defined by \( f\left(x\right)=3x-8 \) . What is the value of \( f\left(7\right) \) ?
原库参考解析
函数记号・代入与复合基础
The function \( f \) is defined by \( f\left(x\right)=10x^2-32x-152 \). What is the value of \( f\left(0\right) \)?
原库参考解析
函数记号・代入与复合挑战
The function \( g \) is defined by \( g\left(x\right)=4x-6 \). What is the value of \( g\left(-7\right) \) ?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) is defined by \( f\left(x\right)=4x^{-1} \) . What is the value of \( f\left(21\right) \) ?
原库参考解析
函数记号・代入与复合挑战
The function p is defined by p(n)=7n3. What is the value of n when p(n) is equal to 56?
原库参考解析
函数记号・代入与复合进阶
The function f(w) = 6w2 gives the area of a rectangle, in square feet (ft2), if its width is w ft and its length is 6 times its width. Which of the following is the best interpretation of f (14) = 1,176?
原库参考解析
函数记号・代入与复合进阶
The total cost f (x), in dollars, to lease a car for 36 months from a particular car dealership is given by f (x) = 36x + 1,000, where x is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is $400?
原库参考解析
