Question

select the correct answer.
which statement describes the end behavior of this absolute value function?
graph of an absolute value function with vertex at (3, 0), y-intercept at (0, 3), and passing through (-3, 6) and (6, 3)
a. as x approaches positive infinity, f(x) approaches positive infinity.
b. as x approaches negative infinity, f(x) approaches negative infinity.
c. as x approaches positive infinity, f(x) approaches negative infinity.
d. as x approaches negative infinity, f(x) is no longer continuous.

Explanation:

To determine the end - behavior of the absolute - value function from its graph:

1. Analyze the right - hand end (as \(x

ightarrow+\infty\)):

• Observe the part of the graph where \(x\) is large and positive. As \(x\) gets larger and larger in the positive direction (approaches positive infinity), we can see that the \(y\) - values (the values of \(f(x)\)) are also getting larger and larger in the positive direction. So, as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\).

1. Analyze the left - hand end (as \(x

ightarrow-\infty\)):

• Look at the part of the graph where \(x\) is large and negative. As \(x\) gets larger and larger in the negative direction (approaches negative infinity), the \(y\) - values (the values of \(f(x)\)) are getting larger and larger in the positive direction (since the left - hand side of the absolute - value graph has a negative slope, but when \(x\) is negative and its magnitude increases, \(y = f(x)\) increases). So, option B is incorrect because \(f(x)\) does not approach negative infinity as \(x

ightarrow-\infty\). Option C is incorrect because as \(x
ightarrow+\infty\), \(f(x)\) approaches positive infinity, not negative infinity. Option D is incorrect because the absolute - value function is continuous everywhere, including as \(x
ightarrow-\infty\).

Answer:

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