find the domain and range of X/x^2+1
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EXPLANATION.
Domain and range :
Equation : (x)/(x² + 1).
As we know that,
Domain : (x)/(x² + 1).
⇒ x² + 1 ≠ 0.
⇒ x² ≠ - 1.
Domain x ∈ R.
Range : (x)/(x² + 1).
We can write equation as,
⇒ y = (x)/(x² + 1).
⇒ y(x² + 1) = (x).
⇒ yx² + y = x.
⇒ yx² - x + y = 0.
As we know that,
x are real then, D ≥ 0.
⇒ b² - 4ac ≥ 0.
⇒ (-1)² - 4(y)(y) ≥ 0.
⇒ 1 - 4y² ≥ 0.
⇒ 4y² - 1 ≤ 0.
⇒ (2y)² - (1)² ≤ 0.
⇒ (2y + 1)(2y - 1) ≤ 0.
Zeroes are,
⇒ 2y + 1 = 0.
⇒ y = - 1/2.
⇒ 2y - 1 = 0.
⇒ 2y = 1.
⇒ y = 1/2.
Put this point on wavy curve method, we get.
Range x ∈ [-1/2, 1/2].
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