Question

as \\(x\\) approaches negative infinity, \\(f(x)\\) approaches negative infinity.
as \\(x\\) approaches positive infinity, \\(f(x)\\) approaches negative infinity.
as \\(x\\) approaches negative infinity, \\(f(x)\\) is no longer continuous.
as \\(x\\) approaches positive infinity, \\(f(x)\\) approaches positive infinity.

Explanation:

🆕 New Concept Discovered: End Behavior of Functions
How the graph behaves at the far left and far right.

Step 1: Analyze the right side of the graph

As we move to the right along the x-axis, \( x \) approaches positive infinity (\( x \to \infty \)).

Looking at the graph, the right arm of the absolute value function points upwards and continues to rise. This means the function value \( f(x) \) increases without bound, approaching positive infinity (\( f(x) \to \infty \)).

Step 2: Analyze the left side of the graph

As we move to the left along the x-axis, \( x \) approaches negative infinity (\( x \to -\infty \)).

Looking at the graph, the left arm of the absolute value function also points upwards and continues to rise. This means the function value \( f(x) \) approaches positive infinity (\( f(x) \to \infty \)).

Step 3: Match with the given options

Let's evaluate the choices:

• Option 1: "As \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity." (Incorrect, it approaches positive infinity)
• Option 2: "As \( x \) approaches positive infinity, \( f(x) \) approaches negative infinity." (Incorrect, it approaches positive infinity)
• Option 3: "As \( x \) approaches negative infinity, \( f(x) \) is no longer continuous." (Incorrect, the function remains continuous)
• Option 4: "As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity." (Correct, as established in Step 1)

Answer:

As x approaches positive infinity, f(x) approaches positive infinity.