Find the range of the following functions f (x) = \(\frac{1}{4-x^2}

1.	Find the range of the following functions f (x) = \(\frac{1}{4-x^2}\)
Answer» Given,f, (x) =\(\frac{1}{4-x^2}\) Let y = f (x) y =\(\frac{1}{4-x^2}\) x² = \(\frac{4y-1}{y}\) x = \(\sqrt{\frac{4y-1}{y}}\) Clearly, x will assume real values, if ⇒ 4y – 1 ≥ 0 and y ≠ 0 4y ≥ 1 and y ≠ 0 ⇒ y ≥ \(\frac{1}{4}\)and y ≠ 0 ⇒ y ∈ (– ∞, 0) ∪ [ \(\frac{1}{1}\) , ∞] ∴ Range of y = (– ∞, 0) ∪ [ \(\frac{1}{1}\) , ∞]

Find the range of the following functions f (x) = \(\frac{1}{4-x^2}\)

Answer»

Given,f, (x) =\(\frac{1}{4-x^2}\)

Let y = f (x) y =\(\frac{1}{4-x^2}\)

x² = \(\frac{4y-1}{y}\)

x = \(\sqrt{\frac{4y-1}{y}}\)

Clearly, x will assume real values, if

⇒ 4y – 1 ≥ 0 and y ≠ 0

4y ≥ 1 and y ≠ 0

⇒ y ≥ \(\frac{1}{4}\)and y ≠ 0

⇒ y ∈ (– ∞, 0) ∪ [ \(\frac{1}{1}\) , ∞]

∴ Range of y = (– ∞, 0) ∪ [ \(\frac{1}{1}\) , ∞]