# Composite Function Calculator

The calculator will find the composition of the functions, with steps shown. It will also evaluate the composition at the specified point, if needed.

A point is 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.

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

## Solution

$$\left(f\circ g\right)\left(x\right) = f{\left(g{\left(x \right)} \right)} = f{\left(x + 7 \right)} = \frac{1}{\color{red}{\left(x + 7\right)}^{2} + \color{red}{\left(x + 7\right)}} = \frac{1}{x^{2} + 15 x + 56}$$$$$\left(g\circ f\right)\left(x\right) = g{\left(f{\left(x \right)} \right)} = g{\left(\frac{1}{x^{2} + x} \right)} = \color{red}{\left(\frac{1}{x^{2} + x}\right)} + 7 = 7 + \frac{1}{x^{2} + x}$$$

$$\left(f\circ f\right)\left(x\right) = f{\left(f{\left(x \right)} \right)} = f{\left(\frac{1}{x^{2} + x} \right)} = \frac{1}{\color{red}{\left(\frac{1}{x^{2} + x}\right)}^{2} + \color{red}{\left(\frac{1}{x^{2} + x}\right)}} = \frac{x^{2} \left(x + 1\right)^{3}}{x \left(x + 1\right)^{2} + x + 1}$$$$$\left(g\circ g\right)\left(x\right) = g{\left(g{\left(x \right)} \right)} = g{\left(x + 7 \right)} = \color{red}{\left(x + 7\right)} + 7 = x + 14$$$

$$\left(f\circ g\right)\left(x\right) = \frac{1}{x^{2} + 15 x + 56}$$$A $$\left(g\circ f\right)\left(x\right) = 7 + \frac{1}{x^{2} + x}$$$A
$$\left(f\circ f\right)\left(x\right) = \frac{x^{2} \left(x + 1\right)^{3}}{x \left(x + 1\right)^{2} + x + 1}$$$A $$\left(g\circ g\right)\left(x\right) = x + 14$$$A