Find the Domain and Range f(x)=x^2-3x+5

$f (x) = x^{2} - 3 x + 5$

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.

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Rewrite the equation in vertex form.

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Complete the square for $x^{2} - 3 x + 5$ .

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

$a = 1, b = - 3, c = 5$

Consider the vertex form of a parabola.

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

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

$d = - \frac{3}{2 (1)}$

Multiply $2$ by $1$ .

$d = - \frac{3}{2}$

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

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Simplify each term.

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Raise $- 3$ to the power of $2$ .

$e = 5 - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 5 - \frac{9}{4}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 5 \cdot \frac{4}{4} - \frac{9}{4}$

Combine $5$ and $\frac{4}{4}$ .

$e = \frac{5 \cdot 4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{5 \cdot 4 - 9}{4}$

Simplify the numerator.

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Multiply $5$ by $4$ .

$e = \frac{20 - 9}{4}$

Subtract $9$ from $20$ .

$e = \frac{11}{4}$

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

${(x - \frac{3}{2})}^{2} + \frac{11}{4}$

Set $y$ equal to the new right side.

$y = {(x - \frac{3}{2})}^{2} + \frac{11}{4}$

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

$a = 1$

$h = \frac{3}{2}$

$k = \frac{11}{4}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{11}{4})$

The range of a parabola that opens up starts at its vertex $(\frac{3}{2}, \frac{11}{4})$ and extends to infinity.

Interval Notation:

$[\frac{11}{4}, \infty)$

Set-Builder Notation:

${y | y \geq \frac{11}{4}}$

Determine the domain and range.

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

Range: $[\frac{11}{4}, \infty), {y | y \geq \frac{11}{4}}$

Find the Domain and Range f(x)=x^2-3x+5

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