Solve y=\left|x^2\right|-\left|x-1\right|

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = {2 x + 1, x < 1 2 x - 1, x > 1

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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 \in (- \infty, 1) \cap x = - \frac{1 + 5 + 4 y}{2} \cup x \in (- \infty, 1) \cap x = \frac{- 1 + 5 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 + - 3 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 - - 3 + 4 y}{2}

Evaluate

y = x^{2} - ∣ x - 1 ∣

When the expression in absolute value bars is not negative, remove the bars

y = x^{2} - ∣ x - 1 ∣

Swap the sides of the equation

x^{2} - ∣ x - 1 ∣ = y

Move the expression to the left side

x^{2} - ∣ x - 1 ∣ - y = 0

Separate the equation into 2 possible cases

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

Solve the equation

More Steps

x = \frac{1 + - 3 + 4 y}{2} x = \frac{1 - - 3 + 4 y}{2}, x - 1 \geq 0 x^{2} - (- (x - 1)) - y = 0, x - 1 < 0

Solve the inequality

More Steps

x = \frac{1 + - 3 + 4 y}{2} x = \frac{1 - - 3 + 4 y}{2}, x \geq 1 x^{2} - (- (x - 1)) - y = 0, x - 1 < 0

Solve the equation

More Steps

x = \frac{1 + - 3 + 4 y}{2} x = \frac{1 - - 3 + 4 y}{2}, x \geq 1 x = \frac{- 1 + 5 + 4 y}{2} x = - \frac{1 + 5 + 4 y}{2}, x - 1 < 0

Solve the inequality

More Steps

x = \frac{1 + - 3 + 4 y}{2} x = \frac{1 - - 3 + 4 y}{2}, x \geq 1 x = \frac{- 1 + 5 + 4 y}{2} x = - \frac{1 + 5 + 4 y}{2}, x < 1

Find the intersection

x \in [1, + \infty) \cap x = \frac{1 + - 3 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 - - 3 + 4 y}{2} x = \frac{- 1 + 5 + 4 y}{2} x = - \frac{1 + 5 + 4 y}{2}, x < 1

Find the intersection

x \in [1, + \infty) \cap x = \frac{1 + - 3 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 - - 3 + 4 y}{2} x \in (- \infty, 1) \cap x = - \frac{1 + 5 + 4 y}{2} \cup x \in (- \infty, 1) \cap x = \frac{- 1 + 5 + 4 y}{2}

Solution

x \in (- \infty, 1) \cap x = - \frac{1 + 5 + 4 y}{2} \cup x \in (- \infty, 1) \cap x = \frac{- 1 + 5 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 + - 3 + 4 y}{2} \cup x \in [1, + \infty) \cap x = \frac{1 - - 3 + 4 y}{2}

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