Question

select the correct answer.

which statement describes the end behavior of the function \\(f(x) = \frac{1}{7}|x - 4| + 3\\)?

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

Explanation:

Identify the function

The given function is:

$$f(x) = \frac{1}{7}|x - 4| + 3$$

Analyze the absolute value term

Using the Range of Absolute Value knowledge point:

$$|x - 4| \ge 0 \quad \text{for all } x \in \mathbb{R}$$

Evaluate limit as x approaches negative infinity

As \(x \to -\infty\), the term inside the absolute value becomes highly negative:

$$x - 4 \to -\infty$$

Taking the absolute value yields:

$$|x - 4| \to +\infty$$

Multiplying by the positive coefficient and adding the constant:

$$f(x) = \frac{1}{7}|x - 4| + 3 \to +\infty$$

Evaluate limit as x approaches positive infinity

As \(x \to +\infty\):

$$x - 4 \to +\infty$$

Taking the absolute value yields:

$$|x - 4| \to +\infty$$

Thus:

$$f(x) = \frac{1}{7}|x - 4| + 3 \to +\infty$$

Match with the given options

The second option states: "As x approaches negative infinity, f(x) approaches positive infinity." This matches our finding.

Answer:

• As x approaches negative infinity, f(x) approaches negative infinity.
• As x approaches negative infinity, f(x) approaches positive infinity. (Correct answer)
• As x approaches positive infinity, f(x) approaches negative infinity.
• As x approaches positive infinity, f(x) is no longer continuous.