Check if the following function have an inverse function. If yes, fin

1.	Check if the following function have an inverse function. If yes, find the inverse function.f(x) = \(\sqrt{4x+5}\) √(4x + 5)
Answer» f(x) = \(\sqrt{4x+5},x\geq\frac{-5}4\) Let f(x₁) = f(x₂) \(\therefore\) \(\sqrt{4x_1+5}=\sqrt{4x_2+5}\) \(\therefore\) x₁ = x₂ ∴ f is a one-one function. f(x) = \(\sqrt{4x+5}=y, \) say(y) \(\geq0\) Squaring on both sides, we get y² = 4x + 5 ∴ x = \(\frac{y^2-5}4\) ∴ For every y we can get x. ∴ f is an onto function. ∴ x = \(\frac{y^2-5}4=f^{-1}(y)\) Replacing y by x, we get f^-1(x) = \(\frac{x^2-5}4\)

Check if the following function have an inverse function. If yes, find the inverse function.f(x) = \(\sqrt{4x+5}\) √(4x + 5)

Answer»

f(x) = \(\sqrt{4x+5},x\geq\frac{-5}4\)

Let f(x₁) = f(x₂)

\(\therefore\) \(\sqrt{4x_1+5}=\sqrt{4x_2+5}\)

\(\therefore\) x₁ = x₂

∴ f is a one-one function.

f(x) = \(\sqrt{4x+5}=y, \) say(y) \(\geq0\)

Squaring on both sides, we get

y² = 4x + 5

∴ x = \(\frac{y^2-5}4\)

∴ For every y we can get x.

∴ f is an onto function.

∴ x = \(\frac{y^2-5}4=f^{-1}(y)\)

Replacing y by x, we get

f^-1(x) = \(\frac{x^2-5}4\)