Questions: Question 10 Which of the following is a one-to-one function? (A) f(x)=(1,4),(2,5),(3,4),(2,7) (B) f(x)=(-1,-1),(-2,-8),(2,-8),(1,1) (C) f(x)=(1,4),(2,5),(3,6),(4,7) (D) f(x)=(-1,1),(-2,4),(2,4),(1,1)

Solution Steps

To determine which of the given functions is one-to-one, we need to check if each input (x-value) maps to a unique output (y-value). A function is one-to-one if no two different inputs map to the same output.

1. For each function, create a set of the output values.
2. Compare the length of the set of output values to the length of the function's list of pairs.
3. If the lengths are equal, the function is one-to-one.

A function \( f \) is one-to-one if and only if \( f(a) \neq f(b) \) for all \( a \neq b \). This means that no two different inputs map to the same output.

We will analyze each function to determine if it is one-to-one.

• Outputs: \( \{4, 5, 4, 7\} \)
• Unique Outputs: \( \{4, 5, 7\} \)
• Since the number of unique outputs (3) is less than the number of pairs (4), function A is not one-to-one.

• Outputs: \( \{-1, -8, -8, 1\} \)
• Unique Outputs: \( \{-1, -8, 1\} \)
• Since the number of unique outputs (3) is less than the number of pairs (4), function B is not one-to-one.

• Outputs: \( \{4, 5, 6, 7\} \)
• Unique Outputs: \( \{4, 5, 6, 7\} \)
• Since the number of unique outputs (4) is equal to the number of pairs (4), function C is one-to-one.

• Outputs: \( \{1, 4, 4, 1\} \)
• Unique Outputs: \( \{1, 4\} \)
• Since the number of unique outputs (2) is less than the number of pairs (4), function D is not one-to-one.

Final Answer