Chapter 7, Problem 28CCSR

Solution

To compare: The end behaviors of the functions $f\left(x\right)=x^3+1$ and $g\left(x\right)=5x^2-7$ .

  $\begin{array}{ccccc}\text{Function}& x& \text{End behavior}& x& \text{End behavior}\\ f\left(x\right)=x^3+1& \infty & \infty & -\infty & \infty \\ g\left(x\right)=5x^2-7& \infty & \infty & -\infty & \infty \end{array}$

Given information:

  $f\left(x\right)=x^3+1$ and $g\left(x\right)=5x^2-7$

Calculation:

The end behaviors of the function depend on their degree.

If degree is odd then it follows. Leading coefficient positive.

  $\begin{array}{l}\text{If }x\to \infty \text{ then }f\left(x\right)\to \infty \\ \text{If }x\to -\infty \text{ then }f\left(x\right)\to -\infty \end{array}$

If degree is even then it follows. Leading coefficient positive.

  $\begin{array}{l}\text{If }x\to \infty \text{ then }f\left(x\right)\to \infty \\ \text{If }x\to -\infty \text{ then }f\left(x\right)\to \infty \end{array}$

Now compare the end behavior of right side $f\left(x\right)=x^3+1$ and $g\left(x\right)=5x^2-7$ as shown in the table.

  $\begin{array}{ccccc}\text{Function}& \text{Degree}& \begin{array}{l}\text{Sign of Leading}\\ \text{Coefficient}\end{array}& x& \text{End behaviour}\\ f\left(x\right)=x^3+1& 3\left(\text{odd}\right)& +& x\to \infty & \infty \\ g\left(x\right)=5x^2-7& 2\left(\text{even}\right)& +& x\to \infty & \infty \end{array}$

End behavior of the left side:

  $\begin{array}{ccccc}\text{Function}& \text{Degree}& \begin{array}{l}\text{Sign of Leading}\\ \text{Coefficient}\end{array}& x& \text{End behaviour}\\ f\left(x\right)=x^3+1& 3\left(\text{odd}\right)& +& x\to -\infty & -\infty \\ g\left(x\right)=5x^2-7& 2\left(\text{even}\right)& +& x\to -\infty & \infty \end{array}$