Algebra Examples

$f (x) = \sqrt{x^{2} + 1} - 2$

Step 1

Set the radicand in $\sqrt{x^{2} + 1}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 1 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$x^{2} \geq - 1$

Step 2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

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:

$[- 1, \infty)$

Set-Builder Notation:

${y | y \geq - 1}$

Step 5

Determine the domain and range.

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

Range: $[- 1, \infty), {y | y \geq - 1}$

Step 6