Question

which statement could be used to explain why \\(f(x) = 2x - 3\\) has an inverse relation that is a function?

• the graph of \\(f(x)\\) passes the vertical line test.
• \\(f(x)\\) is a one-to-one function.
• the graph of the inverse of \\(f(x)\\) passes the horizontal line test.
• \\(f(x)\\) is not a function.

Explanation:

Analyze the definition of an inverse function

For a function \(f(x)\) to have an inverse relation that is also a function, every unique output value \(y\) must correspond to exactly one unique input value \(x\). This characteristic defines a one-to-one function. Using the Inverse Functions concept, we know that if a function is one-to-one, its inverse relation is guaranteed to be a function.

Evaluate the given options

• "The graph of \(f(x)\) passes the vertical line test": This only proves that \(f(x)\) itself is a function, not that its inverse is a function.
• "\(f(x)\) is a one-to-one function": This is the correct definition. If \(f(x)\) is one-to-one, its inverse relation is a function.
• "The graph of the inverse of \(f(x)\) passes the horizontal line test": For the inverse to be a function, the graph of the original function \(f(x)\) must pass the horizontal line test, or the graph of the inverse must pass the vertical line test.
• "\(f(x)\) is not a function": This is false, as \(f(x) = 2x - 3\) is a linear function.

Answer:

• The graph of \(f(x)\) passes the vertical line test.
• \(f(x)\) is a one-to-one function. (Correct answer)
• The graph of the inverse of \(f(x)\) passes the horizontal line test.
• \(f(x)\) is not a function.