Trigonometry Examples

$g (x) = \frac{2 x^{2} - 3 x - 5}{x^{2} + 1}$

Step 1

Set the denominator in $\frac{2 x^{2} - 3 x - 5}{x^{2} + 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{3 + \sqrt{58}}{2}, - \frac{3 - \sqrt{58}}{2}]$

Set-Builder Notation:

${y | - \frac{3 + \sqrt{58}}{2} \leq y \leq - \frac{3 - \sqrt{58}}{2}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[- \frac{3 + \sqrt{58}}{2}, - \frac{3 - \sqrt{58}}{2}], {y | - \frac{3 + \sqrt{58}}{2} \leq y \leq - \frac{3 - \sqrt{58}}{2}}$

Step 6