What is the answer to i(x)= 1/(x+1) ?

The answer to i(x)= 1/(x+1) is x=0.30024…-0.62481…i,x=-1.30024…+0.62481…i

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i(x)= 1/(x+1)

i (x) = \frac{1}{x + 1}

x = 0.30024 \dots - 0.62481 \dots i, x = - 1.30024 \dots + 0.62481 \dots i

Solution steps

i (x) = \frac{1}{x + 1}

Substitute

x = a + bi

i (a + bi) = \frac{1}{a + bi + 1}

Multiply both sides by

a + 1 + ib

i (a + bi) (a + 1 + ib) = \frac{1}{a + bi + 1} (a + 1 + ib)

Expand

(- 2 ab - b) + i (a + a^{2} - b^{2}) = 1

Rewrite

1

in standard complex form:

1 + 0 i

(- 2 ab - b) + i (a + a^{2} - b^{2}) = 1 + 0 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[- 2 ab - b = 1 a + a^{2} - b^{2} = 0]

[- 2 ab - b = 1 a + a^{2} - b^{2} = 0] : (a = 0.30024 \dots, a = - 1.30024 \dots, b = - 0.62481 \dots b = 0.62481 \dots)

Substitute back

x = a + bi

x = 0.30024 \dots - 0.62481 \dots i, x = - 1.30024 \dots + 0.62481 \dots i

\frac{z + 1}{z - 1} = 2 + i

z^{2} - z - 1 - 3 i = 0

\frac{3 - a + 2 b}{5} - 3 i + a + bi = 4 - i

(7 - 7 i) - 8 = x - (2 + y i)

What is the answer to i(x)= 1/(x+1) ?
The answer to i(x)= 1/(x+1) is x=0.30024…-0.62481…i,x=-1.30024…+0.62481…i