Chapter 2.1, Problem 5DC

Solution

(a.)

To Make: A conjecture about the end behavior of the given family of functions.

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to \infty$ when $x\to \infty$ and $f\left(x\right)\to -\infty$ when $x\to -\infty$ .

Given:

The family of functions, $f\left(x\right)=x^n$ , where $n$ is odd.

Concept used:

Even exponential power of a negative quantity is positive and odd exponential power of a negative quantity is negative.

Calculation:

The given family of functions is, $f\left(x\right)=x^n$ , where $n$ is odd.

Since the exponential power is odd, for positive $x$ , $f\left(x\right)=x^n$ is positive and for negative $x$ , $f\left(x\right)=x^n$ is negative.

Thus, the end behavior of this family of functions can be given as:

  $f\left(x\right)\to \infty$ when $x\to \infty$ and $f\left(x\right)\to -\infty$ when $x\to -\infty$ .

Conclusion:

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to \infty$ when $x\to \infty$ and $f\left(x\right)\to -\infty$ when $x\to -\infty$ .

(b.)

To Make: A conjecture about the end behavior of the given family of functions.

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to -\infty$ when $x\to \infty$ and $f\left(x\right)\to \infty$ when $x\to -\infty$ .

Given:

The family of functions, $f\left(x\right)=-x^n$ , where $n$ is odd.

Concept used:

Even exponential power of a negative quantity is positive and odd exponential power of a negative quantity is negative.

Calculation:

The given family of functions is, $f\left(x\right)=-x^n$ , where $n$ is odd.

Since the exponential power is odd, for positive $x$ , $x^n$ is positive and hence $f\left(x\right)=-x^n$ is negative and for negative $x$ , $x^n$ is negative and hence $f\left(x\right)=-x^n$ is positive.

Thus, the end behavior of this family of functions can be given as:

  $f\left(x\right)\to -\infty$ when $x\to \infty$ and $f\left(x\right)\to \infty$ when $x\to -\infty$ .

Conclusion:

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to -\infty$ when $x\to \infty$ and $f\left(x\right)\to \infty$ when $x\to -\infty$ .

(c.)

To Make: A conjecture about the end behavior of the given family of functions.

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to \infty$ when $x\to \pm \infty$ .

Given:

The family of functions, $f\left(x\right)=x^n$ , where $n$ is even.

Concept used:

Even exponential power of a negative quantity is positive and odd exponential power of a negative quantity is negative.

Calculation:

The given family of functions is, $f\left(x\right)=x^n$ , where $n$ is even.

Since the exponential power is even, for both positive and negative $x$ , $f\left(x\right)=x^n$ is positive.

Thus, the end behavior of this family of functions can be given as:

  $f\left(x\right)\to \infty$ when $x\to \pm \infty$ .

Conclusion:

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to \infty$ when $x\to \pm \infty$ .

(d.)

To Make: A conjecture about the end behavior of the given family of functions.

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to -\infty$ when $x\to \pm \infty$ .

Given:

The family of functions, $f\left(x\right)=-x^n$ , where $n$ is even.

Concept used:

Even exponential power of a negative quantity is positive and odd exponential power of a negative quantity is negative.

Calculation:

The given family of functions is, $f\left(x\right)=-x^n$ , where $n$ is even.

Since the exponential power is even, for both positive and negative $x$ , $x^n$ is positive and hence $f\left(x\right)=-x^n$ is negative.

Thus, the end behavior of this family of functions can be given as:

  $f\left(x\right)\to -\infty$ when $x\to \pm \infty$ .

Conclusion:

It has been determined that the end behavior of the given family of functions is given as,

  $f\left(x\right)\to -\infty$ when $x\to \pm \infty$ .