Question

which of the following function types exhibit the end behavior $f(x) \to \infty$ as $x \to \infty$ and $f(x) \to \infty$ as $x \to -\infty$? select all that apply. (1 point) \square power, $y = x^n$; $n$ is even and greater than zero. \square reciprocal, $y = \frac{1}{x}$ \square root, $y = \sqrtn{x}$; $n$ is even and greater than zero. \square identity, $y = x$ \square exponential, $y = b^x$, $b > 0$ \square absolute value, $y = |x|$

Explanation:

Step1: Analyze power function

For $y=x^n$, $n$ even, $n>0$:
As $x\to\infty$, $x^n\to\infty$; as $x\to-\infty$, $(-x)^n=x^n\to\infty$.

Step2: Analyze reciprocal function

For $y=\frac{1}{x}$:
As $x\to\infty$, $\frac{1}{x}\to0$; as $x\to-\infty$, $\frac{1}{x}\to0$.

Step3: Analyze even root function

For $y=\sqrt[n]{x}$, $n$ even, $n>0$:
Domain is $x\geq0$, so $x\to-\infty$ is undefined.

Step4: Analyze identity function

For $y=x$:
As $x\to\infty$, $x\to\infty$; as $x\to-\infty$, $x\to-\infty$.

Step5: Analyze exponential function

For $y=b^x$, $b>0$:
If $01$, as $x\to\infty$, $b^x\to\infty$; as $x\to-\infty$, $b^x\to0$.

Step6: Analyze absolute value function

For $y=|x|$:
As $x\to\infty$, $|x|\to\infty$; as $x\to-\infty$, $|x|=|-x|\to\infty$.

Answer:

power, $y=x^n$; $n$ is even and greater than zero
absolute value, $y=|x|$