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・代入与复合挑战
The graph of a polynomial function \( y=f(x) \) in the xy-plane passes through the point (7, 6). What is the value of \( f(7) \)?
参考答案
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=3x-8 \). What is the value of \( f(9) \)?
参考答案
函数记号・代入与复合进阶
Maya is enlarging the area of a photograph that measures 7 centimeters by 5 centimeters. The resulting area, in square centimeters, of the photograph is given by the function \( S(x)=(7x)(5x) \) , where x represents the scale factor used to increase both the length and width of the photograph. Which statement is the best interpretation of\( S(3)=315 \)?
参考答案
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 $800, how many car stickers must they sell?
参考答案
函数记号・代入与复合进阶
The function k is defined by \( k(x)=x^3+80 \) . What is the value of \( k(x) \) when \( x=2 \)?
参考答案
函数记号・代入与复合挑战
The function \( f \) is defined by \( f(x)=-\frac{1}{4}(x-13)+4 \). For what value of x does \( f(x)=4a \), where a is a constant?
参考答案
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=\frac{4}{5}x-\frac{7}{5} \). For what value of x does \( f(x)=5 \)?
参考答案
函数记号・代入与复合挑战
The function A defined by \( A(x)=x(x+2) \) gives the area, in square inches, for a parallelogram with a height of x inches and a base that is 2 inches longer than its height. Which of the following is the best interpretation of A(50) = 2,600 in this context?
参考答案
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=3x+4 \). What is the value of \( f(x) \) when x = 100?
参考答案
函数记号・代入与复合基础
The function \( f \) is defined by \( f(x)=9x^3+6 \). What is the value of \( f(2) \)?
参考答案
函数记号・代入与复合进阶
The graph of the polynomial function \( f \) in the xy-plane, where \( y=f(x) \), passes through the point (2, 7). What is the value of \( f(2) \)?
参考答案
函数记号・代入与复合进阶
\( f(x)=x+\frac{7}{12} \)
The function \( f \) is defined by the given equation. What is the value of \( f(x) \) when \( x=\frac{5}{12} \)?
The function \( f \) is defined by the given equation. What is the value of \( f(x) \) when \( x=\frac{5}{12} \)?
参考答案
