Graph f(x)=2- square root of x+1

$f (x) = 2 - \sqrt{x + 1}$

Step 1

Find the domain for $y = 2 - \sqrt{x + 1}$ 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 $\sqrt{x + 1}$ greater than or equal to $0$ to find where the expression is defined.

$x + 1 \geq 0$

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

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

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Step 2

To find the radical expression end point, substitute the $x$ value $- 1$ , which is the least value in the domain, into $f (x) = 2 - \sqrt{x + 1}$ .

Replace the variable $x$ with $- 1$ in the expression.

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

Simplify the result.

Simplify each term.

Add $- 1$ and $1$ .

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

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

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

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

$f (- 1) = 2 - 0$

Multiply $- 1$ by $0$ .

$f (- 1) = 2 + 0$

Add $2$ and $0$ .

$f (- 1) = 2$

The final answer is $2$ .

$2$

Step 3

The radical expression end point is $(- 1, 2)$ .

$(- 1, 2)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

Substitute the $x$ value $0$ into $f (x) = 2 - \sqrt{x + 1}$ . In this case, the point is $(0, 1)$ .

Replace the variable $x$ with $0$ in the expression.

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

Simplify the result.

Simplify each term.

Add $0$ and $1$ .

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

Any root of $1$ is $1$ .

$f (0) = 2 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (0) = 2 - 1$

Subtract $1$ from $2$ .

$f (0) = 1$

The final answer is $1$ .

$y = 1$

Substitute the $x$ value $1$ into $f (x) = 2 - \sqrt{x + 1}$ . In this case, the point is $(1, 2 - \sqrt{2})$ .

Replace the variable $x$ with $1$ in the expression.

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

Simplify the result.

Add $1$ and $1$ .

$f (1) = 2 - \sqrt{2}$

The final answer is $2 - \sqrt{2}$ .

$y = 2 - \sqrt{2}$

The square root can be graphed using the points around the vertex $(- 1, 2), (0, 1), (1, 0.59)$

$\begin{array}{c} x & y \\ - 1 & 2 \\ 0 & 1 \\ 1 & 0.59 \end{array}$

Step 5

Graph f(x)=2- square root of x+1

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