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 \( h(x)=4x−9 \). What is the value of \( h(−3) \) ?
原库参考解析
函数记号・代入与复合进阶
If \( f(x)=x+3 \) and \( g(x)=3x \) , what is the value of \( 6f(3)−g(3) \) ?
原库参考解析
函数记号・代入与复合挑战
The function \( g \) is defined by \( g\left(x\right)=7x^{-1} \). What is the value of \( g\left(8\right) \)?
原库参考解析
函数记号・代入与复合挑战
The kinetic energy, in joules, of an object with mass \( 41 \) kilograms traveling at a speed of \( v \) meters per second is given by the function \( K \), where \( K\left(v\right)=\frac{1}{2}\left(41\right)\left(v^2\right) \). Which of the following is the best interpretation of \( K\left(0\right)=0 \)in this context?
原库参考解析
函数记号・代入与复合进阶
The function \( g \) is defined by \( g\left(x\right)=7x^{-1} \). What is the value of \( g\left(13\right) \)?
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 \( f\left(x\right)=10x+4 \). 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
\( m(t)=−0.0274(\frac{t}{7})^{2}+7.3873(\frac{t}{7})+75.032 \)
The function m gives the predicted body mass \( m(t) \) in kilograms (kg), of a male giraffe \( t \) days after it was born in a wildlife reserve, where \( t\le390 \). Which of the following is the best interpretation of the statement "\( m(370) \) is approximately equal to \( 389 \)"in this context?
原库参考解析
函数记号・代入与复合挑战
\( m\left(t\right)=-0.0274\left(\frac{t}{7}\right)^2+7.3873\left(\frac{t}{7}\right)+75.032 \)
The function m gives the predicted body mass m(t), in kilograms (kg), of a male giraffe t days after it was born in a wildlife reserve, where \( t\le390 \). Which of the following is the best interpretation of the statement “ m(140) is approximately equal to 212” in this context?
原库参考解析
函数记号・代入与复合进阶
The function \( f(x)=180(x-2) \) 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 \( g(x)=18-x \) . What is the value of \( g(3) \)?
原库参考解析
函数记号・代入与复合进阶
\( f(x)=x^3+8x+17 \)
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?
原库参考解析
函数记号・代入与复合挑战
The function \( f \) is defined by \( f(x)=\frac{x+16}{5} \) , and \( f(a)=-19 \), where a is a constant. What is the value of a?
原库参考解析
