Question

which statement best describes the function? a) the function is onto. b) the function is one-to-one. c) the function is onto and one-to-one. d) the function is neither one-to-one nor onto.

Explanation:

Step1: Recall definitions

• One - to - one (injective): A function \(y = f(x)\) is one - to - one if no two different \(x\) values have the same \(y\) value. We 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.
• Onto (surjective): A function \(y=f(x)\) is onto if for every \(y\) in the codomain, there exists an \(x\) in the domain such that \(f(x)=y\).

Step2: Analyze the graph for one - to - one

Looking at the graph, we apply the horizontal line test. As we move a horizontal line across the graph, we see that no horizontal line intersects the graph more than once. So the function passes the horizontal line test, which means it is one - to - one.

Step3: Analyze the graph for onto

The graph of the function seems to cover all real - valued \(y\) values (since it goes from \(-\infty\) to \(+\infty\) in the \(y\) - direction). For any real number \(y\), we can find an \(x\) such that \(f(x) = y\). So the function is onto.

Since the function is both one - to - one and onto, we can conclude the correct option.

Answer:

C) The function is onto and one - to - one.