Algebra Examples

$y = \frac{1}{1 + x^{2}}$

Step 1

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

$1 + x^{2} = 0$

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:

$(0, 1]$

Set-Builder Notation:

${y | 0 < y \leq 1}$

Step 5

Determine the domain and range.

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

Range: $(0, 1], {y | 0 < y \leq 1}$

Step 6