Chapter 4.1, Problem 24E

To determine

Write the type, degree, and leading coefficient of the polynomial function. Also describe the end behavior of the function.

Answer to Problem 24E

Degree of the polynomial function is 0.

Leading coefficient will be $13$ .

Given function is a constant polynomial.

End behavior is $f\left(x\right)\to \infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to +\infty$ .

Explanation of Solution

Given information:

A function is given as

  $f\left(x\right)=13$

Concept used:

A polynomial function is of the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.

Leading coefficient of a polynomial function is the coefficient of the leading term.

Degree of the polynomial is the degree of leading term or the height degree in the polynomial function.

For the polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$ the leading coefficient is $a_n$ and the degree is $n$ .

The end behavior can describe the graph of a polynomial function as  $x$  approaches $+\infty$ or as  $x$  approaches $-\infty$ .

The end behavior of a polynomial function can be determined by the leading coefficient and the degree of the polynomial.

If degree is even and leading coefficient is negative.

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

If degree is odd and leading coefficient is negative.

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

If degree is odd and leading coefficient is positive.

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

If degree is even and leading coefficient is positive.

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

Calculation:

Consider the given function.

  $f\left(x\right)=13$

Above function can be written as

  $f\left(x\right)=13x^0$

Now, degree of the term is a whole number and coefficient is real.

So, the function is polynomial function.

Now, degree of the polynomial function is 0.

Leading term is $13x^0$ .

So, leading coefficient will be $13$ .

As, degree of the polynomial function is 0, so it is a constant polynomial.

As leading coefficient is positive and degree is even,

So, end behavior is $f\left(x\right)\to \infty$ as $x\to -\infty$ and $f\left(x\right)\to \infty$ as $x\to +\infty$ .