Solve y=|x|/(x^2 1)

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = {\frac{1}{x ^{2}}, x < 0 - \frac{1}{x ^{2}}, x > 0

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Solve for x

Solve for y

x = 0 x \in (- \infty, 0) \cap x = - \frac{1}{y} x \in [0, + \infty) \cap x = \frac{1}{y}

Evaluate

y = \frac{∣ x ∣}{x ^{2} \times 1}

Any expression multiplied by 1 remains the same

y = \frac{∣ x ∣}{x ^{2}}

Swap the sides of the equation

\frac{∣ x ∣}{x ^{2}} = y

Cross multiply

∣ x ∣ = x^{2} y

Simplify the equation

∣ x ∣ = y x^{2}

Rewrite the expression

∣ x ∣ - y x^{2} = 0

Separate the equation into 2 possible cases

x - y x^{2} = 0, x \geq 0 - x - y x^{2} = 0, x < 0

Solve the equation

More Steps

x = 0 x = \frac{1}{y}, x \geq 0 - x - y x^{2} = 0, x < 0

Solve the equation

More Steps

x = 0 x = \frac{1}{y}, x \geq 0 x = 0 x = - \frac{1}{y}, x < 0

Find the intersection

x = 0 x \in [0, + \infty) \cap x = \frac{1}{y} x = 0 x = - \frac{1}{y}, x < 0

Find the intersection

x = 0 x \in [0, + \infty) \cap x = \frac{1}{y} x \in (- \infty, 0) \cap x = - \frac{1}{y}

Solution

x = 0 x \in (- \infty, 0) \cap x = - \frac{1}{y} x \in [0, + \infty) \cap x = \frac{1}{y}

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