Question

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:

Step1: Analyze the right end (as \( x \to \infty \))

Look at the graph's right - hand side. As \( x \) moves towards positive infinity (\( x \to \infty \)), the graph of \( y = f(x) \) is rising upwards, which means \( f(x)\to\infty \) as \( x \to \infty \).

Step2: Analyze the left end (as \( x \to -\infty \))

Look at the graph's left - hand side. As \( x \) moves towards negative infinity (\( x \to -\infty \)), the graph of \( y = f(x) \) is falling downwards, which means \( f(x)\to-\infty \) as \( x \to -\infty \).

Answer:

As \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \) (the corresponding option among the choices, for example, if the third option is "as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \)", then the answer is that option).