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 \( f(x)=5x^3+6 \). What is the value of \( f(2) \)?
参考答案
函数记号・代入与复合进阶
\( f(x)=x^3+6x+11 \)
For the given function \( f \), the graph of \( y=f(x) \) in the xy-plane passes through the point (0, b), where b is a constant. What is the value of b?
For the given function \( f \), the graph of \( y=f(x) \) in the xy-plane passes through the point (0, b), where b is a constant. What is the value of b?
参考答案
函数记号・代入与复合挑战
The function \( f \) is defined by \( f(x)=7x-205 \). If the point (k, 2k) lies on the graph of \( y=f(x) \) in the xy-plane, where k is a constant, what is the value of k?
参考答案
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 2.75 inches, it has an estimated breaking strength of 9,528.75 pounds. What is the estimated breaking strength, in pounds, of a rope of this type that has a circumference of 3.50 inches?
参考答案
函数记号・代入与复合挑战
The function m is defined by \( m(x)=3x+2 \), and the function p is defined by \( p(x)=2-x \). What is the value of \( 4m(2)−p(2) \)?
参考答案
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=6x+5 \) . 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 h is defined by \( h(x)=x(x^2+29) \). What is the value of \( h(10) \)?
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 \( p\left(x\right)=5x-190 \) to determine their profit p(x), in dollars, for selling x car stickers. In order to earn a profit of $400, 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 \( f\left(x\right)=180\left(x-2\right) \) gives the sum of the interior angles, in degrees, for a polygon with x sides. What is the sum of the interior angles, in degrees, for a polygon with 35 sides?
原库参考解析
函数记号・代入与复合挑战
The function g is defined by \( g\left(x\right)=18-x \). What is the value of \( g\left(3\right) \)?
原库参考解析
函数记号・代入与复合进阶
\( f\left(x\right)=7x+800 \)
The function shown gives the total cost f(x), in dollars, for making x units of a certain product. What is the total cost, in dollars, for making 100 units of this product?
原库参考解析
函数记号・代入与复合进阶
The function \( f \) is defined by \( f(x)=5x^{2}+x \) . What is the value of \( f\left(6\right) \)?
原库参考解析
