Chapter 9.CT, Problem 1CT

To determine

The domain of the given function

Answer to Problem 1CT

Solution:

The domain of the given function is $\left\{x|x\ne \frac{1}{2},x\ne -4\right\}$ or $\left(-\infty ,-4\right)\cup \left(-4,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$.

Explanation of Solution

Given:

The given function is,

$f\left(x\right)=\frac{-3}{2x^2+7x-4}$

Calculation:

Consider the given function,

$f\left(x\right)=\frac{-3}{2x^2+7x-4}$

Now, equate the denominator with zero,

$2x^2+7x-4=0$

Factorize the above equation.

$\begin{array}{rll}2x^2+\left(8x-x\right)-4& =& 0\\ 2x^2+8x-x-4& =& 0\\ 2x\left(x+4\right)-1\left(x+4\right)& =& 0\\ \left(2x-1\right)\left(x+4\right)& =& 0\end{array}$

Solve the first factor for x.

$\begin{array}{rll}\left(2x-1\right)& =& 0\\ 2x& =& 1\\ x& =& \frac{1}{2}\end{array}$

Solve the second factor for x.

$\begin{array}{rll}\left(x+4\right)& =& 0\\ x& =& -4\end{array}$

Therefore, the domain of the given function is $\left\{x|x\ne \frac{1}{2},x\ne -4\right\}$ or $\left(-\infty ,-4\right)\cup \left(-4,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$.

Conclusion:

Thus, the domain of the given function is $\left\{x|x\ne \frac{1}{2},x\ne -4\right\}$ or $\left(-\infty ,-4\right)\cup \left(-4,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$.