Find the Domain and Range f(x) = fourth root of 3x^2+1

$f (x) = \sqrt[4]{3 x^{2} + 1}$

Set the radicand in $\sqrt[4]{3 x^{2} + 1}$ greater than or equal to $0$ to find where the expression is defined.

$3 x^{2} + 1 \geq 0$

Solve for $x$ .

Subtract $1$ from both sides of the inequality.

$3 x^{2} \geq - 1$

Divide each term in $3 x^{2} \geq - 1$ by $3$ and simplify.

Divide each term in $3 x^{2} \geq - 1$ by $3$ .

$\frac{3 x^{2}}{3} \geq \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} \geq \frac{- 1}{3}$

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} \geq - \frac{1}{3}$

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

All real numbers

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

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

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${y | y \geq 1}$

Determine the domain and range.

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

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

Find the Domain and Range f(x) = fourth root of 3x^2+1

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