Given : f(x) = x^2 and g(x) = 2^x
The inverse of g is a function, but the inverse of f is not a function. Explain why the statement is true.
the inverse of f will have two values for each value of indepentdent variable. the inverse of g does not.
Thank you so much!!! :)
To determine if the inverse of a function exists, we need to check if the function is one-to-one, meaning that each input value corresponds to a unique output value.
In the case of g(x) = 2^x, we can see that for every value of x, we get a unique value of g(x). This is because exponential functions, like 2^x, have a one-to-one relationship. Therefore, g(x) is one-to-one, and its inverse exists as a function.
Now let's consider f(x) = x^2. For this function, as we input positive and negative values of x, we get different outputs. However, when we take the square root of the outputs, we get two possible solutions -- one positive and one negative. This means that for a single output value, there are two possible input values. Hence, f(x) is not one-to-one, making it impossible to define its inverse as a function.
To summarize, the inverse of g(x) = 2^x exists as a function because it has a one-to-one relationship. On the other hand, the inverse of f(x) = x^2 does not exist as a function because it is not one-to-one.