Solve for f (complex solution)
f=\frac{1}{x^{2}+6x+7}
x\neq \sqrt{2}-3\text{ and }x\neq -\left(\sqrt{2}+3\right)
Solve for f
f=\frac{1}{x^{2}+6x+7}
x\neq \sqrt{2}-3\text{ and }x\neq -\sqrt{2}-3
Solve for x (complex solution)
x=\sqrt{2+\frac{1}{f}}-3
x=-\sqrt{2+\frac{1}{f}}-3\text{, }f\neq 0
Solve for x
\left\{\begin{matrix}x=-\sqrt{2+\frac{1}{f}}-3\text{; }x=\sqrt{2+\frac{1}{f}}-3\text{, }&f>0\\x=-\sqrt{2+\frac{1}{f}}-3\text{; }x=\sqrt{2+\frac{1}{f}}-3\text{, }&f\leq -\frac{1}{2}\end{matrix}\right.
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f^{-1}=x^{2}+6x+9-2
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
f^{-1}=x^{2}+6x+7
Subtract 2 from 9 to get 7.
\frac{1}{f}=x^{2}+6x+7
Reorder the terms.
1=fx^{2}+6xf+f\times 7
Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.
fx^{2}+6xf+f\times 7=1
Swap sides so that all variable terms are on the left hand side.
\left(x^{2}+6x+7\right)f=1
Combine all terms containing f.
\frac{\left(x^{2}+6x+7\right)f}{x^{2}+6x+7}=\frac{1}{x^{2}+6x+7}
Divide both sides by x^{2}+6x+7.
f=\frac{1}{x^{2}+6x+7}
Dividing by x^{2}+6x+7 undoes the multiplication by x^{2}+6x+7.
f=\frac{1}{x^{2}+6x+7}\text{, }f\neq 0
Variable f cannot be equal to 0.
f^{-1}=x^{2}+6x+9-2
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.
f^{-1}=x^{2}+6x+7
Subtract 2 from 9 to get 7.
\frac{1}{f}=x^{2}+6x+7
Reorder the terms.
1=fx^{2}+6xf+f\times 7
Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.
fx^{2}+6xf+f\times 7=1
Swap sides so that all variable terms are on the left hand side.
\left(x^{2}+6x+7\right)f=1
Combine all terms containing f.
\frac{\left(x^{2}+6x+7\right)f}{x^{2}+6x+7}=\frac{1}{x^{2}+6x+7}
Divide both sides by x^{2}+6x+7.
f=\frac{1}{x^{2}+6x+7}
Dividing by x^{2}+6x+7 undoes the multiplication by x^{2}+6x+7.
f=\frac{1}{x^{2}+6x+7}\text{, }f\neq 0
Variable f cannot be equal to 0.
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Limits
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