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Question
f^{-1}z=e^{z}+fz
Solve the equation
\begin{align}&f=\frac{-e^{z}+\sqrt{e^{2z}+4z^{2}}}{2z}\\&f=-\frac{e^{z}+\sqrt{e^{2z}+4z^{2}}}{2z}\end{align}
Evaluate
f^{-1}z=e^{z}+fz
Rewrite the expression
zf^{-1}=e^{z}+zf
Rewrite the expression
More Steps
Evaluate
zf^{-1}
\text{Express with a positive exponent using }a^{-n}=\frac{1}{a^n}
z\times \frac{1}{f}
Rewrite the expression
\frac{z}{f}
\frac{z}{f}=e^{z}+zf
Multiply both sides of the equation by LCD
\frac{z}{f}\times f=\left(e^{z}+zf\right)f
Simplify the equation
z=\left(e^{z}+zf\right)f
Simplify the equation
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Evaluate
\left(e^{z}+zf\right)f
Apply the distributive property
e^{z}f+zf\times f
Calculate
More Steps
Multiply the terms
f\times f
Calculate
f^{1+1}
Calculate
f^{2}
e^{z}f+zf^{2}
z=e^{z}f+zf^{2}
Swap the sides of the equation
e^{z}f+zf^{2}=z
Move the expression to the left side
e^{z}f+zf^{2}-z=0
Rewrite in standard form
zf^{2}+e^{z}f-z=0
\text{Substitute a=}z\text{,b=}e^{z}\text{ and c=}-z\text{ into the quadratic formula }f\text{=}\frac{-b\pm\sqrt{b^2-4ac}}{2a}
f=\frac{-e^{z}\pm \sqrt{\left(e^{z}\right)^{2}-4z\left(-z\right)}}{2z}
Simplify the expression
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Evaluate
\left(e^{z}\right)^{2}-4z\left(-z\right)
Evaluate the power
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Evaluate
\left(e^{z}\right)^{2}
Transform the expression
e^{z\times 2}
Use the commutative property to reorder the terms
e^{2z}
e^{2z}-4z\left(-z\right)
Multiply
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Multiply the terms
4z\left(-z\right)
Rewrite the expression
-4z\times z
Multiply the terms
-4z^{2}
e^{2z}-\left(-4z^{2}\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
e^{2z}+4z^{2}
f=\frac{-e^{z}\pm \sqrt{e^{2z}+4z^{2}}}{2z}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&f=\frac{-e^{z}+\sqrt{e^{2z}+4z^{2}}}{2z}\\&f=\frac{-e^{z}-\sqrt{e^{2z}+4z^{2}}}{2z}\end{align}
Solution
\begin{align}&f=\frac{-e^{z}+\sqrt{e^{2z}+4z^{2}}}{2z}\\&f=-\frac{e^{z}+\sqrt{e^{2z}+4z^{2}}}{2z}\end{align}
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