Determine whether the function is one-to-one. If it is, find a formula for its inverse. [tex]f(x)=x^3-6[/tex]
Is the function one-to-one?
Yes
No
Answer (2)
Determine that f ( x ) = x 3 − 6 is one-to-one by analyzing its derivative, f ′ ( x ) = 3 x 2 , which is always non-negative.
Set y = x 3 − 6 and solve for x in terms of y , obtaining x = 3 y + 6 .
Swap x and y to express the inverse function: f − 1 ( x ) = 3 x + 6 .
The function is one-to-one, and its inverse is f − 1 ( x ) = 3 x + 6 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 − 6 and asked to determine if it is one-to-one and, if so, find its inverse. A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once. Alternatively, a function is one-to-one if it is strictly increasing or strictly decreasing.
Checking if the Function is One-to-One To determine if the function is one-to-one, we can analyze its derivative. The derivative of f ( x ) = x 3 − 6 is f ′ ( x ) = 3 x 2 . Since x 2 is always non-negative, 3 x 2 is also always non-negative. This means that the function is always increasing. However, f ′ ( x ) = 0 only at x = 0 , so the function is strictly increasing. Therefore, the function is one-to-one.
Finding the Inverse Function To find the inverse of the function, we set y = f ( x ) = x 3 − 6 and solve for x in terms of y . y = x 3 − 6 Add 6 to both sides: y + 6 = x 3 Take the cube root of both sides: x = 3 y + 6
Stating the Inverse Function Now, we replace y with x to express the inverse function as f − 1 ( x ) . f − 1 ( x ) = 3 x + 6 Thus, the inverse of the function f ( x ) = x 3 − 6 is f − 1 ( x ) = 3 x + 6 .
Examples
One-to-one functions are essential in cryptography, where each input must correspond to a unique output to ensure secure encoding and decoding of messages. For example, if f ( x ) = x 3 − 6 is used to encode a message, the inverse function f − 1 ( x ) = 3 x + 6 is needed to decode it. This ensures that the original message can be recovered accurately. Understanding inverse functions helps in designing secure communication systems and data encryption methods.
The function f ( x ) = x 3 − 6 is one-to-one because its derivative is strictly positive, indicating it is strictly increasing. The inverse of the function can be found as f − 1 ( x ) = 3 x + 6 . Therefore, the answer is Yes, the function is one-to-one.
;
