Question

For a function to have an inverse function, it must be a one-to-one function, which means it must pass the horizontal line test. In other words, the function must strictly increase or strictly decrease over its entire domain.

So, how can you approach a function that isn't one-to-one? For instance, how could you determine an inverse function for the function f(x) = x²? If you were to somehow change the function, how would that affect the resulting inverse function?

Answer 1

For functions that aren't one-to-one, such as f(x) = x², an inverse can be found by restricting the domain to ensure the function passes the horizontal line test. By restricting the domain to 0 ≤ x ≤ 20, it becomes strictly increasing, and its inverse would then be the square root function over that range. The inverse function is applicable only within the range of the modified function.

Step-by-step explanation:

To approach a function that isn't one-to-one, such as f(x) = x², you need to consider its domain. The function f(x) = x² does not pass the horizontal line test over all real numbers, as it is not a strict increase or decrease, which means it doesn't have an inverse function across its entire domain. However, by restricting the domain, for instance to 0 ≤ x ≤ 20, you can ensure that it does pass the horizontal line test and can define an inverse function.

For the modified function where the domain is restricted, the function increases for all x in the domain, making it one-to-one. The resulting inverse is g(y) = √ y where y is the output of the function f(x). It's important to note that the inverse function will only work for the values of y within the range of the modified function.

When working with triangle side lengths, for example, applying the Pythagorean Theorem reveals the need to 'undo' the square operation. You can obtain the side length 'a' by taking the square root of c² - b², effectively inverting the square function.