Write the domain of the function in interval notation. Write numbers as integers or simplified fractions.

Ask by Vaughan Thornton. in the United States

Mar 22,2025

Question

Write the domain of the function in interval notation. Write numbers as integers or simplified fractions. $$ r(x)=\frac{2 x-5}{2 x^{2}+1 x-6} $$

UpStudy AI · Accepted Answer

The function is given by
$$
r(x)=\frac{2x-5}{2x^2+x-6}.
$$
To find the domain, we must exclude values of $$ x $$ that make the denominator equal to zero.

1. Set the denominator equal to zero:
   $$
   2x^2+x-6=0.
   $$

2. Solve the quadratic equation using the quadratic formula:
   $$
   x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},
   $$
   where $$ a=2 $$, $$ b=1 $$, and $$ c=-6 $$.

3. Substitute the values into the formula:
   $$
   x=\frac{-1\pm\sqrt{1^2-4(2)(-6)}}{2(2)}=\frac{-1\pm\sqrt{1+48}}{4}=\frac{-1\pm\sqrt{49}}{4}.
   $$

4. Since $$\sqrt{49} = 7$$:
   $$
   x=\frac{-1+7}{4}=\frac{6}{4}=\frac{3}{2} \quad \text{or} \quad x=\frac{-1-7}{4}=\frac{-8}{4}=-2.
   $$

5. These values are not in the domain. Therefore, the domain of $$ r(x) $$ is all real numbers except $$ x=-2 $$ and $$ x=\frac{3}{2} $$.

In interval notation, the domain is:
$$
(-\infty,-2)\cup(-2,\frac{3}{2})\cup(\frac{3}{2},\infty).
$$
The domain of $$ r(x) $$ is all real numbers except $$ x = -2 $$ and $$ x = \frac{3}{2} $$. In interval notation, this is: $$ (-\infty, -2) \cup \left(-2, \frac{3}{2}\right) \cup \left(\frac{3}{2}, \infty\right) $$

Write the domain of the function in interval notation. Write numbers as integers or simplified fractions.

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions