sketch a graph whose domain is the interval [0, infinity) but whose domain must be restricted to [0,4] so that the function can have an inverse. (The only one i can think of is a parabola but she said we cant use that one)
Consider the 2 statements:
I. if f(a)=b then f(inverse)(b)=a
II. if f(inverse)(b)=a then f(a)=b
Which statement is NOT necessarily true if f(x) does not pass the horizontal line test and WHY?
To sketch a graph that meets the given requirements, we will consider the function f(x) = √(x) on the domain [0, infinity). However, we need to restrict the domain to [0,4] to ensure that the function has an inverse.
To understand the significance of the domain restriction, we can analyze the concept of inverse functions. For a function to have an inverse, it must satisfy both the one-to-one (injective) and onto (surjective) properties. In simple terms, this means that each input value should correspond to a unique output value, and every output value should have a corresponding input value.
Now, let's address the second part of your question regarding the statements I. and II.:
Statement I. states that if f(a) = b, then f(inverse)(b) = a. This statement is true by definition for any function and its inverse. If a particular input value, a, maps to an output value, b, then the inverse function will map b back to a.
Statement II. states that if f(inverse)(b) = a, then f(a) = b. This statement is also true for any function and its inverse. If the inverse function maps b back to a, then the original function should map a back to b.
Therefore, both statements I. and II. are generally true for any function and its inverse, irrespective of whether the original function passes the horizontal line test. The horizontal line test is only applicable to determine whether a function has an inverse that is also a function.
In conclusion, if the function f(x) does not pass the horizontal line test, it does not mean that one of the statements I. or II. will be false. The horizontal line test primarily determines if the inverse of the function is also a function or if it is a relation.