复合函数计算器
逐步求解函数的复合
该计算器将求出函数 $$$f(x)$$$ 和 $$$g(x)$$$ 的复合 $$$(f\circ g)(x)$$$、$$$(g\circ f)(x)$$$、$$$(f\circ f)(x)$$$ 和 $$$(f\circ g)(x)$$$，并显示步骤。如有需要，还会在指定点对这些复合进行求值。
相关计算器: 函数运算计算器
您的输入
求$$$f{\left(x \right)} = \frac{1}{x^{2} + x}$$$与$$$g{\left(x \right)} = x + 7$$$的复合函数。
解答
$$$\left(f\circ g\right)\left(x\right) = f\left(g\left(x\right)\right) = f\left(x + 7\right) = \frac{1}{{\color{red}\left(x + 7\right)}^{2} + {\color{red}\left(x + 7\right)}} = \frac{1}{\left(x + 7\right) \left(x + 8\right)}$$$
$$$\left(g\circ f\right)\left(x\right) = g\left(f\left(x\right)\right) = g\left(\frac{1}{x^{2} + x}\right) = {\color{red}\left(\frac{1}{x^{2} + x}\right)} + 7 = 7 + \frac{1}{x^{2} + x}$$$
$$$\left(f\circ f\right)\left(x\right) = f\left(f\left(x\right)\right) = f\left(\frac{1}{x^{2} + x}\right) = \frac{1}{{\color{red}\left(\frac{1}{x^{2} + x}\right)}^{2} + {\color{red}\left(\frac{1}{x^{2} + x}\right)}} = \frac{x^{2} \left(x + 1\right)^{2}}{x^{2} + x + 1}$$$
$$$\left(g\circ g\right)\left(x\right) = g\left(g\left(x\right)\right) = g\left(x + 7\right) = {\color{red}\left(x + 7\right)} + 7 = x + 14$$$
答案
$$$\left(f\circ g\right)\left(x\right) = \frac{1}{\left(x + 7\right) \left(x + 8\right)}$$$A
$$$\left(g\circ f\right)\left(x\right) = 7 + \frac{1}{x^{2} + x}$$$A
$$$\left(f\circ f\right)\left(x\right) = \frac{x^{2} \left(x + 1\right)^{2}}{x^{2} + x + 1}$$$A
$$$\left(g\circ g\right)\left(x\right) = x + 14$$$A