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) = 9x3 + 8
Answer» f(x) = 9x³ + 8 Let f(x₁ ) = f(x₂) \(\therefore9x^3_1+8=9x^3_2+8\) \(\therefore\) x₁ = x₂ ∴ f is a one-one function. ∴ f(x) = 9x³ + 8 = y, (say) \(\therefore\) x = \(\sqrt[3]{\frac{y-8}9}\) ∴ For every y we can get x. ∴ f is an onto function. \(\therefore\) x = \(\sqrt[3]{\frac{y-8}9}\) = f^-1(y) Replacing y by x, we get f^-1(x) = \(\sqrt[3]{\frac{y-8}9}\)

Check if the following function have an inverse function. If yes, find the inverse function.f(x) = 9x3 + 8

Answer»

f(x) = 9x³ + 8

Let f(x₁ ) = f(x₂)

\(\therefore9x^3_1+8=9x^3_2+8\)

\(\therefore\) x₁ = x₂

∴ f is a one-one function.

∴ f(x) = 9x³ + 8 = y, (say)

\(\therefore\) x = \(\sqrt[3]{\frac{y-8}9}\)

∴ For every y we can get x.

∴ f is an onto function.

\(\therefore\) x = \(\sqrt[3]{\frac{y-8}9}\) = f^-1(y)

Replacing y by x, we get

f^-1(x) = \(\sqrt[3]{\frac{y-8}9}\)