# 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.

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

## Solution

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

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

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

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

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