Find the Domain and Range f(x)=x^2+3

$f (x) = x^{2} + 3$

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:

${x | x \in ℝ}$

Locate the vertex of the parabola.

Tap for more steps…

Rewrite the equation in vertex form.

Tap for more steps…

Complete the square for $x^{2} + 3$ .

Tap for more steps…

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1, b = 0, c = 3$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 (1)}$

Cancel the common factor of $0$ and $2$ .

Tap for more steps…

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (1)}$

Cancel the common factors.

Tap for more steps…

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{0}{1}$

Divide $0$ by $1$ .

$d = 0$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps…

Simplify each term.

Tap for more steps…

Raising $0$ to any positive power yields $0$ .

$e = 3 - \frac{0}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 3 - \frac{0}{4}$

Divide $0$ by $4$ .

$e = 3 - 0$

Multiply $- 1$ by $0$ .

$e = 3 + 0$

Add $3$ and $0$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

${(x + 0)}^{2} + 3$

Set $y$ equal to the new right side.

$y = {(x + 0)}^{2} + 3$

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = 0$

$k = 3$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(0, 3)$

The range of a parabola that opens up starts at its vertex $(0, 3)$ and extends to infinity.

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${y | y \geq 3}$

Determine the domain and range.

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

Range: $[3, \infty), {y | y \geq 3}$

Find the Domain and Range f(x)=x^2+3

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store