Graph f(x) = square root of -(x+3)+1

$f (x) = \sqrt{- (x + 3)} + 1$

Find the domain for $y = \sqrt{- (x + 3)} + 1$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Set the radicand in $\sqrt{- (x + 3)}$ greater than or equal to $0$ to find where the expression is defined.

$- (x + 3) \geq 0$

Solve for $x$ .

Simplify $- (x + 3)$ .

Apply the distributive property.

$- x - 1 \cdot 3 \geq 0$

Multiply $- 1$ by $3$ .

$- x - 3 \geq 0$

Add $3$ to both sides of the inequality.

$- x \geq 3$

Divide each term in $- x \geq 3$ by $- 1$ and simplify.

Divide each term in $- x \geq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{3}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{3}{- 1}$

Simplify the right side.

Divide $3$ by $- 1$ .

$x \leq - 3$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 3]$

Set-Builder Notation:

${x | x \leq - 3}$

Interval Notation:

$(- \infty, - 3]$

Set-Builder Notation:

${x | x \leq - 3}$

To find the radical expression end point, substitute the $x$ value $- 3$ , which is the least value in the domain, into $f (x) = \sqrt{- (x + 3)} + 1$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \sqrt{- ((- 3) + 3)} + 1$

Simplify the result.

Simplify each term.

Add $- 3$ and $3$ .

$f (- 3) = \sqrt{- 0} + 1$

Multiply $- 1$ by $0$ .

$f (- 3) = \sqrt{0} + 1$

Rewrite $0$ as $0^{2}$ .

$f (- 3) = \sqrt{0^{2}} + 1$

Pull terms out from under the radical, assuming positive real numbers.

$f (- 3) = 0 + 1$

Add $0$ and $1$ .

$f (- 3) = 1$

The final answer is $1$ .

$1$

The radical expression end point is $(- 3, 1)$ .

$(- 3, 1)$

The square root can be graphed using the points around the vertex $(- 3, 1),$

$\begin{array}{c} x & y \\ - 3 & 1 \end{array}$

Graph f(x) = square root of -(x+3)+1

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