Question

Find the inverse of

Answer 1

Answer:

$\sf f^{-1}(x) = \pm \sqrt{\dfrac{x + 3}{7}}$

Step-by-step explanation:

To find the inverse of the given function $\sf y = 7x^2 - 3$, we'll swap the roles of $\sf x$ and $\sf y$ and solve for the new $\sf y$.

$\sf y = 7x^2 - 3$

Swap $\sf x$ and $\sf y$:

$\sf x = 7y^2 - 3$

Solve for $\sf y$:

$\sf x + 3 = 7y^2$

$\sf 7y^2 = x + 3$

$\sf y^2 = \dfrac{x + 3}{7}$

$\sf y = \pm \sqrt{\dfrac{x + 3}{7}}$

So, the inverse function is:

$\sf f^{-1}(x) = \pm \sqrt{\dfrac{x + 3}{7}}$

It's important to note that we take the positive and negative square root because a function and its inverse are reflections across the line $\sf y = x$, and both branches of the square root are needed to capture this reflection.