Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = 36 + x ^ { 2 } , x \leq 0$
A)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \geq 36$
B)No, $f$ does not have an inverse function.
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x + 36 } , x \geq 36$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = \sqrt { x - 36 } , x \geq 36$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \leq 36$

Question

Quizplus · Accepted Answer

Given $f(x) = 36 + x^2$ for $x \leq 0$, the function is one-to-one and thus has an inverse. To find the inverse, solve for $x$ in terms of $y$, leading to $x = -\sqrt{y - 36}$, which is defined for $y \geq 36$.

Determine Whether the Function Has an Inverse Function, If It f(x)=36+x2,x≤0f ( x ) = 36 + x ^ { 2 } , x \leq 0f(x)=36+x2,x≤0

Determine Whether the Function Has an Inverse Function, If It $f ( x ) = 36 + x ^ { 2 } , x \leq 0$