Consider the following relations. Which is a one-to-one function?

• A. $\left\lbrace{(1,4),(2,5),(3,6),(4,7)}\right\rbrace$
• B. $\left\lbrace{(1,2),(1,3),(2,4)}\right\rbrace$
• C. ${\left\lbrace(2,3),(3,3),(4,5)\right\rbrace}$
• D. ${\left\lbrace(1,1),(2,2),(3,1),(9,9)\right\rbrace}$

Answer: (A)

Verified step by step guidance

1. Understand that a one-to-one function (also called an injective function) is a function where each output value corresponds to exactly one input value. This means no two different inputs produce the same output.
2. To determine if a function is one-to-one, you can use the Horizontal Line Test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.
3. Alternatively, algebraically test if the function is one-to-one by assuming \( f(a) = f(b) \) and then showing that this implies \( a = b \). This confirms that different inputs cannot produce the same output.
4. If the function is given explicitly, set \( f(a) = f(b) \) and solve the resulting equation for \( a \) and \( b \). If the only solution is \( a = b \), the function is one-to-one.
5. If the function passes the one-to-one test, it has an inverse function. Understanding this concept is key to working with one-to-one functions in algebra.