Question

Finding all inverses Find all the inverses associated with the following functions, and state their domains.ƒ(x) = x² -2x + 6

Accepted Answer

Step 1: Determine if the function is one-to-one.\u003e To find the inverse of a function, it must be one-to-one. The function $$ f(x) = x^2 - 2x + 6  is a $$quadratic function, which is not one-to-one over all real numbers. Therefore, we need to restrict the domain to make it one-to-one. Quadratic functions are one-to-one on intervals where they are either strictly increasing or strictly decreasing. Since the parabola opens upwards, we can consider the vertex to find the interval. The vertex form of a quadratic function is $$ f(x) = a(x-h)^2 + k $$, where $$ (h, k)  is $$the vertex. Step 2: Find the vertex of the quadratic function.\u003e The vertex of $$ f(x) = x^2 - 2x + 6 $$ can be found using the formula $$ h = -\frac{b}{2a} . $$Here, $$ a = 1 $$ and $$ b = -2 $$, so $$ h = -\frac{-2}{2 	imes 1} = 1 . $$Substitute $$ x = 1 $$ into the function to find $$ k $$: $$ f(1) = 1^2 - 2 	imes 1 + 6 = 5 . $$Thus, the vertex is $$ (1, 5) . $$Step 3: Restrict the domain.\u003e To make the function one-to-one, restrict the domain to either $$ x \leq 1  or$$ $$ x \geq 1 . $$This will ensure the function is either strictly increasing or strictly decreasing. For this problem, let's choose $$ x \geq 1 $$ for simplicity, where the function is increasing. Step 4: Solve for the inverse function.\u003e To find the inverse, swap $$ x $$ and $$ y  in $$the equation $$ y = x^2 - 2x + 6 $$ and solve for $$ y . $$This gives $$ x = y^2 - 2y + 6 . $$Rearrange to form a quadratic equation in terms of $$ y $$: $$ y^2 - 2y + (6 - x) = 0 . $$Use the quadratic formula $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}  to $$solve for $$ y . $$Here, $$ a = 1 $$, $$ b = -2 $$, and $$ c = 6 - x . $$Step 5: State the inverse function and its domain.\u003e The inverse function will be $$ f^{-1}(x) = 1 + \sqrt{x - 5} $$ for $$ x \geq 5 $$, since we chose the domain $$ x \geq 1 $$ for the original function. The domain of the inverse function is determined by the range of the original function, which is $$ y \geq 5 .$$

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = x² -2x + 6

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Key Concepts

Inverse Functions

Domain and Range

Completing the Square

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