Composite Function Calculator

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

Optional.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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