Composite Function Calculator

Find the composition of functions step by step

The calculator will find the compositions $$$(f\circ g)(x)$$$, $$$(g\circ f)(x)$$$, $$$(f\circ f)(x)$$$, and $$$(f\circ g)(x)$$$ of the functions $$$f(x)$$$ and $$$g(x)$$$, with steps shown. It will also evaluate the compositions at the specified point if needed.

Related calculator: Operations on Functions Calculator

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Your Input

Find the composition of $$$f{\left(x \right)} = \frac{1}{x^{2} + x}$$$ and $$$g{\left(x \right)} = x + 7$$$.

Solution

$$$\left(f\circ g\right)\left(x\right) = f\left(g\left(x\right)\right) = f\left(x + 7\right) = \frac{1}{{\color{red}\left(x + 7\right)}^{2} + {\color{red}\left(x + 7\right)}} = \frac{1}{\left(x + 7\right) \left(x + 8\right)}$$$

$$$\left(g\circ f\right)\left(x\right) = g\left(f\left(x\right)\right) = g\left(\frac{1}{x^{2} + x}\right) = {\color{red}\left(\frac{1}{x^{2} + x}\right)} + 7 = 7 + \frac{1}{x^{2} + x}$$$

$$$\left(f\circ f\right)\left(x\right) = f\left(f\left(x\right)\right) = f\left(\frac{1}{x^{2} + x}\right) = \frac{1}{{\color{red}\left(\frac{1}{x^{2} + x}\right)}^{2} + {\color{red}\left(\frac{1}{x^{2} + x}\right)}} = \frac{x^{2} \left(x + 1\right)^{2}}{x^{2} + x + 1}$$$

$$$\left(g\circ g\right)\left(x\right) = g\left(g\left(x\right)\right) = g\left(x + 7\right) = {\color{red}\left(x + 7\right)} + 7 = x + 14$$$

Answer

$$$\left(f\circ g\right)\left(x\right) = \frac{1}{\left(x + 7\right) \left(x + 8\right)}$$$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