Given the function [tex]$f(x)=-2(x+5)^2+6$[/tex]:
a) Determine the equation of the inverse of [tex]$f(x)$[/tex]. Note: do not use the quadratic formula.
b) Is the inverse of [tex]$f(x)$[/tex] a function? If not, restrict the domain of [tex]$f(x)$[/tex] such that its inverse is a function. Include all possibilities.

Find the inverse by swapping x and y in the original equation: x = − 2 ( y + 5 ) 2 + 6 .
Solve for y to get the inverse function: f − 1 ( x ) = − 5 ± 2 6 − x ​ ​ .
Determine that the inverse is not a function due to the ± sign.
Restrict the domain of f ( x ) to x ≥ − 5 or x ≤ − 5 to make the inverse a function: f − 1 ( x ) = − 5 + 2 6 − x ​ ​ or f − 1 ( x ) = − 5 − 2 6 − x ​ ​ .
f − 1 ( x ) = − 5 ± 2 6 − x ​ ​ ; not a function, restrict domain to x ≥ − 5 or x ≤ − 5 ​

Explanation

Problem Analysis We are given the function f ( x ) = − 2 ( x + 5 ) 2 + 6 and we need to find its inverse. Then, we need to determine if the inverse is a function. If it is not a function, we need to restrict the domain of the original function so that the inverse is a function.

Finding the Inverse First, let's find the inverse of the function. To do this, we replace f ( x ) with y , so we have y = − 2 ( x + 5 ) 2 + 6 . Then we swap x and y to get x = − 2 ( y + 5 ) 2 + 6 . Now we solve for y .

Isolating the Square Term Subtract 6 from both sides: x − 6 = − 2 ( y + 5 ) 2 .

Dividing by -2 Divide both sides by -2: − 2 x − 6 ​ = ( y + 5 ) 2 , which simplifies to 2 6 − x ​ = ( y + 5 ) 2 .

Taking the Square Root Take the square root of both sides: ± 2 6 − x ​ ​ = y + 5 .

The Inverse Function Subtract 5 from both sides: y = − 5 ± 2 6 − x ​ ​ . So the inverse function is f − 1 ( x ) = − 5 ± 2 6 − x ​ ​ .

Is the Inverse a Function? Now, let's determine if the inverse is a function. Because of the ± sign, for a single value of x , there are two possible values of y . This means the inverse is not a function.

Restricting the Domain To restrict the domain of f ( x ) such that its inverse is a function, we can restrict f ( x ) to x ≥ − 5 or x ≤ − 5 . This will make the original function one-to-one, and thus its inverse will be a function.

Two Possible Inverses If we restrict the domain of f ( x ) to x ≥ − 5 , then the inverse function is f − 1 ( x ) = − 5 + 2 6 − x ​ ​ . If we restrict the domain of f ( x ) to x ≤ − 5 , then the inverse function is f − 1 ( x ) = − 5 − 2 6 − x ​ ​ .

Domain of the Inverse The domain of the inverse function is determined by the expression inside the square root: 2 6 − x ​ ≥ 0 , which means 6 − x ≥ 0 , so x ≤ 6 . Therefore, the domain of the inverse is x ≤ 6 . The range of f ( x ) is ( − ∞ , 6 ] .

Final Answer Therefore, the equation of the inverse is f − 1 ( x ) = − 5 ± 2 6 − x ​ ​ . The inverse is not a function. To make the inverse a function, we can restrict the domain of f ( x ) to x ≥ − 5 or x ≤ − 5 . If x ≥ − 5 , then f − 1 ( x ) = − 5 + 2 6 − x ​ ​ . If x ≤ − 5 , then f − 1 ( x ) = − 5 − 2 6 − x ​ ​ .


Examples
Understanding inverse functions is crucial in many real-world applications. For example, in cryptography, inverse functions are used to decode messages. If a function encodes a message, the inverse function decodes it back to its original form. Similarly, in computer graphics, inverse functions can be used to transform objects back to their original positions after a series of transformations. This concept is also used in economics to analyze supply and demand curves, where one curve can be seen as the inverse of the other.

Answered by GinnyAnswer | 2025-07-04