Chapter 6, Problem 17MCQ

To determine

Find the domain and range of the given function.

Answer to Problem 17MCQ

The domain and range of the given function is $-\infty <f\left(x\right)<\infty$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^3-2x^2-3x$ .

Calculation:

Draw a table for the values of $f\left(x\right)$ with respect to $x$ .

$x$	$f\left(x\right)=x^3-2x^2-3x$	$\left(x,f\left(x\right)\right)$
$\begin{array}{l}-2\\ -1\\ -0.535\\ 1.868\\ 3\\ 4\end{array}$	$\begin{array}{l}-10\\ 0\\ 0.879\\ -6.06\\ 0\\ 20\end{array}$	$\begin{array}{l}\left(-2,-10\right)\\ \left(-1,0\right)\\ \left(-0.535,0.879\right)\\ \left(1.868,-6.06\right)\\ \left(3,0\right)\\ \left(4,20\right)\end{array}$

The graph of the function is given below.

  

The domain of the function is all real numbers. The end behavior is the behavior of the graph of $f\left(x\right)$ as $x$ approaches positive infinity $\left(x\to +\infty \right)$ or negative infinity $\left(x\to -\infty \right)$ . The degree and leading coefficient of a polynomial function determine the end behavior of the graph and the range of the function.

The domain of the function is all real numbers.

The degree of the polynomial is odd. The range of polynomial with odd degree is all the real numbers.

Hence the domain and range of the given function is $-\infty <f\left(x\right)<\infty$ .