Questions: Determine from the graph of the function below if the function is one-to-one. f(x) = 2x + 3, x ≤ -4 x + 1, x > -4

Solution Steps

To determine if a function is one-to-one, we need to check if it passes the horizontal line test. This means that no horizontal line should intersect the graph of the function more than once. For piecewise functions, we need to check each piece separately and ensure continuity at the boundary points.

A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. This means that for any two different inputs \( x_1 \) and \( x_2 \), the outputs \( f(x_1) \) and \( f(x_2) \) are different, i.e., \( f(x_1) \neq f(x_2) \).

The function given is:

\[ f(x) = \begin{cases} 2x + 3, & x \leq -4 \\ x + 1, & x > -4 \end{cases} \]

This is a piecewise function with two linear components. We need to analyze each piece separately to determine if the function is one-to-one.

1. For \( x \leq -4 \): The function is \( f(x) = 2x + 3 \).

   • This is a linear function with a slope of 2. Linear functions with non-zero slopes are one-to-one because they pass the horizontal line test.

2. For \( x > -4 \): The function is \( f(x) = x + 1 \).

   • This is also a linear function with a slope of 1. Similarly, linear functions with non-zero slopes are one-to-one.

• At \( x = -4 \), the value of the first piece is \( f(-4) = 2(-4) + 3 = -8 + 3 = -5 \).
• The second piece is not defined at \( x = -4 \), so there is no overlap or conflict at this point.

Since both pieces of the function are linear with non-zero slopes and there is no overlap or conflict at the transition point \( x = -4 \), the function is one-to-one.

Final Answer