Question

Find all the inverses associated with the following functions and state their domains. $$f(x)=(x-4)^{2}$$

Short answer

Answer: The inverse of the given function is \(f^{-1}(x) = \sqrt{x} + 4\), and its domain is \(x \geq 0\) or in interval notation, \([0, \infty)\).

Step-by-step solution

Swap x and y

Replace \(f(x)\) with \(y\), then swap the positions of \(x\) and \(y\): $$y=(x-4)^{2}$$ becomes: $$x=(y-4)^{2}$$

Solve for y

Solve the equation for \(y\): Take the square root of both sides: $$\sqrt{x}=y-4$$ Now, isolate \(y\): $$y=\sqrt{x}+4$$

Inverse Function

Now, we have the inverse function: $$f^{-1}(x)=\sqrt{x}+4$$

Determine the Domain

Since the inverse function is a square root, we know that the domain of the function must satisfy the condition \(x\geq 0\). So, the domain of \(f^{-1}(x)\) is: $$x\geq 0$$ or in interval notation: $$[0, \infty)$$