Find the Domain f(x) = natural log of 3/(x^2+1)

$f (x) = \ln (\frac{3}{x^{2} + 1})$

Step 1

Set the argument in $\ln (\frac{3}{x^{2} + 1})$ greater than $0$ to find where the expression is defined.

$\frac{3}{x^{2} + 1} > 0$

Step 2

Solve for $x$ .

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} + 1 = 0$

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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

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

$x = i$

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

$x = - i$

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

$x = i, - i$

The leading coefficient cannot be determined because $\frac{3}{x^{2} + 1}$ is not a polynomial.

Not a polynomial

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $\frac{3}{x^{2} + 1}$ is always greater than $0$ .

All real numbers

Step 3

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

$x^{2} + 1 = 0$

Step 4

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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

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

$x = i$

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

$x = - i$

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

$x = i, - i$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6

Find the Domain f(x) = natural log of 3/(x^2+1)

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