Question

the graph of ( y = f(x) ) is graphed below. what is the end behavior of ( f(x) )?
answer
( circ ) as ( x \to infty ), ( f(x) \to infty ) and as ( x \to -infty ), ( f(x) \to infty )
( circ ) as ( x \to infty ), ( f(x) \to -infty ) and as ( x \to -infty ), ( f(x) \to -infty )
( circ ) as ( x \to infty ), ( f(x) \to infty ) and as ( x \to -infty ), ( f(x) \to -infty )
( circ ) as ( x \to infty ), ( f(x) \to -infty ) and as ( x \to -infty ), ( f(x) \to infty )

Explanation:

To determine the end - behavior of the function \(y = f(x)\) from its graph, we look at the direction of the graph as \(x\) approaches positive infinity (\(x
ightarrow\infty\)) and negative infinity (\(x
ightarrow-\infty\)).

• As \(x

ightarrow\infty\): We observe the right - hand end of the graph. The right - most part of the graph is going downwards, which means as \(x\) gets larger and larger (approaches \(\infty\)), the value of \(f(x)\) approaches \(-\infty\) (i.e., \(f(x)
ightarrow-\infty\)).

• As \(x

ightarrow-\infty\): We observe the left - hand end of the graph. The left - most part of the graph is also going downwards, which means as \(x\) gets smaller and smaller (approaches \(-\infty\)), the value of \(f(x)\) approaches \(-\infty\) (i.e., \(f(x)
ightarrow-\infty\)).

Among the given options, the option that states "as \(x
ightarrow\infty,f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty,f(x)
ightarrow-\infty\)" matches our observation.

Answer:

as \(x \to \infty, f(x) \to -\infty\) and as \(x \to -\infty, f(x) \to -\infty\) (the option with this description)