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 g is defined by\( g\left(x\right)=\sqrt{3x+7} \). What is the value of \( g\left(6\right) \) ?
原库参考解析
函数记号・代入与复合进阶
A student council group is selling car stickers for a fundraiser. They use the function p(x) = 5x -210 to determine their profit p(x), in dollars, for selling x car stickers. In order to earn a profit of $900 , how many car stickers must they sell ?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f\left(x\right)=4^{\left(x-2\right)}+8 \). What is the value of \( f\left(2\right) \)?
原库参考解析
函数记号・代入与复合进阶
The function j is defined by \( j\left(x\right)=\frac{x}{8} \). What is the value of \( j\left(48\right) \)?
原库参考解析
函数记号・代入与复合挑战
The function g is defined by \( g\left(x\right)=-18-9x \). For what value of x does \( g\left(x\right)=-216 \)?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by\( f\left(x\right)=3\left(\frac{1}{4}-x\right)^2+\frac{5}{4} \). What is the value of\( f\left(\frac{1}{4}\right) \) ?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f\left(x\right)=-3x+20 \). What is the value of \( f\left(x\right) \) when \( x=-7 \) ?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) gives the product of a number, x, and a number that is 77 more than x. Which equation defines ?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) is defined by \( f\left(x\right)=3x^2 \). What is the value of \( f\left(7\right) \)?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f\left(x\right)=5x^3+3 \). What is the value of \( f\left(2\right) \)?
原库参考解析
函数记号・代入与复合挑战
For a certain type of rope, the equation \( y=900ax^2 \) , where a is a constant, gives the estimated breaking strength y , in pounds, of a rope with a circumference of x inches. Based on this equation, if a rope of this type has a circumference of 0.75 inches, it has an estimated breaking strength of 708.75 pounds. What is the estimated breaking strength, in pounds, of a rope of this type that has a circumference of 4.50 inches?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f\left(x\right)=7+\sqrt{x} \). What is the value of \( f\left(36\right) \)?
原库参考解析
