Composite Function Calculator

Find the composition of functions step by step

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

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Your Input

Find the composition of f(x)=1x2+xf{\left(x \right)} = \frac{1}{x^{2} + x} and g(x)=x+7g{\left(x \right)} = x + 7.

Solution

(fg)(x)=f(g(x))=f(x+7)=1(x+7)2+(x+7)=1(x+7)(x+8)\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)}

(gf)(x)=g(f(x))=g(1x2+x)=(1x2+x)+7=7+1x2+x\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}

(ff)(x)=f(f(x))=f(1x2+x)=1(1x2+x)2+(1x2+x)=x2(x+1)2x2+x+1\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}

(gg)(x)=g(g(x))=g(x+7)=(x+7)+7=x+14\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

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

(gf)(x)=7+1x2+x\left(g\circ f\right)\left(x\right) = 7 + \frac{1}{x^{2} + x}A

(ff)(x)=x2(x+1)2x2+x+1\left(f\circ f\right)\left(x\right) = \frac{x^{2} \left(x + 1\right)^{2}}{x^{2} + x + 1}A

(gg)(x)=x+14\left(g\circ g\right)\left(x\right) = x + 14A