Find a formula for the inverse.
[tex]f(x)=5 x^3-6[/tex]
[tex]f^{-1}(x)=[/tex]
(Simplify your answer. Type an exact answer, using radicals as needed.)
Answer (1)
Replace f ( x ) with y : y = 5 x 3 − 6 .
Swap x and y : x = 5 y 3 − 6 .
Solve for y : y = 3 5 x + 6 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 3 5 x + 6 .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x 3 − 6 and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.
Replace f(x) with y To find the inverse, we first replace f ( x ) with y , so we have y = 5 x 3 − 6 .
Swap x and y Next, we swap x and y to get x = 5 y 3 − 6 . This is the key step in finding the inverse function.
Add 6 to both sides Now, we solve for y in terms of x . First, add 6 to both sides of the equation: x + 6 = 5 y 3
Divide by 5 Then, divide both sides by 5: 5 x + 6 = y 3
Take the cube root Finally, take the cube root of both sides to isolate y :
y = 3 5 x + 6
Write the inverse function Replace y with f − 1 ( x ) to denote the inverse function: f − 1 ( x ) = 3 5 x + 6
Final Answer Therefore, the inverse function is f − 1 ( x ) = 3 5 x + 6 .
Examples
Imagine you're baking a cake, and f ( x ) represents the process of adding ingredients and baking to get the final cake. The inverse function, f − 1 ( x ) , would be like 'unbaking' the cake to get back the original ingredients. In real life, inverse functions are used in cryptography to decode messages, in physics to reverse processes, and in computer graphics to transform images back to their original state. Understanding inverse functions helps us reverse engineer processes and solve problems in various fields.
