Find the Domain and Range f(x)=9-(x-1)^2

$f (x) = 9 - {(x - 1)}^{2}$

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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Reorder $9$ and $- {(x - 1)}^{2}$ .

$y = - {(x - 1)}^{2} + 9$

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

$a = - 1$

$h = 1$

$k = 9$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(1, 9)$

The range of a parabola that opens down starts at its vertex $(1, 9)$ and extends to negative infinity.

Interval Notation:

$(- \infty, 9]$

Set-Builder Notation:

${y | y \leq 9}$

Determine the domain and range.

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

Range: $(- \infty, 9], {y | y \leq 9}$

Find the Domain and Range f(x)=9-(x-1)^2

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