Which of the following functions has an inverse that is not a function?
a. (1,1),(1,2),(3,4)
b. (1,2),(1,3),(4,3)
c. (1,4),(4,5),(5,6)
d. (0,0),(1,1),(2,2)
b. (1,2),(1,3),(4,3)
To determine which of the given functions has an inverse that is not a function, we need to check if each function satisfies the criteria for inverses.
For a function to have an inverse, it must meet two conditions:
1. It must be a one-to-one function, meaning that no two input values produce the same output value.
2. The function should be onto, meaning that every element in the range has a corresponding element in the domain.
Let's analyze each function:
a. (1,1),(1,2),(3,4)
This function does not satisfy the criteria for a one-to-one function since the input value 1 is associated with two different output values (1 and 2). Therefore, it does not have an inverse that is a function.
b. (1,2),(1,3),(4,3)
Similar to the previous function, this function is not one-to-one because the input value 1 is associated with two different output values (2 and 3). Hence, it does not have an inverse that is a function.
c. (1,4),(4,5),(5,6)
This function satisfies the conditions for a one-to-one function. Each input value is associated with a unique output value, and there are no repeating input values. Therefore, it has an inverse that is a function.
d. (0,0),(1,1),(2,2)
This function is one-to-one since no two input values produce the same output value. It is also onto since every element in the range has a corresponding element in the domain. Thus, it has an inverse that is a function.
Therefore, the only function that does not have an inverse that is a function is option a.
To determine which of the given functions has an inverse that is not a function, we can use the definition of inverse functions.
For a function to have an inverse, each element in the domain must have a unique element in the range, and vice versa. In other words, a function has an inverse if and only if it is a one-to-one (injective) function.
Let's analyze each option:
a. (1,1),(1,2),(3,4)
In this function, the element 1 in the domain maps to two different elements (1 and 2) in the range. Therefore, it does not satisfy the definition of a one-to-one (injective) function. Hence, the inverse is not a function.
b. (1,2),(1,3),(4,3)
Similar to option a, this function also maps the element 1 in the domain to two different elements (2 and 3) in the range. Thus, it is not a one-to-one (injective) function, and its inverse is not a function.
c. (1,4),(4,5),(5,6)
For this function, each element in the domain is mapped to a unique element in the range, and vice versa. So, it is a one-to-one (injective) function, and its inverse will also be a function.
d. (0,0),(1,1),(2,2)
This function maps each element in the domain to the same element in the range. While it is a function, it is not one-to-one (injective) since different elements in the domain are mapped to the same element in the range. Therefore, its inverse is not a function.
In conclusion, options a and b have inverses that are not functions.