Algebra Examples

$y = \sqrt{1 - x} + 2$

Step 1

Reorder $1$ and $- x$ .

$y = \sqrt{- x + 1} + 2$

Step 2

Find the domain for $y = \sqrt{1 - x} + 2$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 2.1

Set the radicand in $\sqrt{- x + 1}$ greater than or equal to $0$ to find where the expression is defined.

$- x + 1 \geq 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the inequality.

$- x \geq - 1$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 1$ by $- 1$ .

$x \leq 1$

Step 2.3

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

Interval Notation:

$(- \infty, 1]$

Set-Builder Notation:

${x | x \leq 1}$

Interval Notation:

$(- \infty, 1]$

Set-Builder Notation:

${x | x \leq 1}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 1$ by $1$ .

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

Step 3.2.1.2

Add $- 1$ and $1$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

$f (1) = 0 + 2$

Step 3.2.2

Add $0$ and $2$ .

$f (1) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 4

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

$(1, 2)$

Step 5

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.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.2.1.2

Add $1$ and $1$ .

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

Step 5.1.2.2

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

$y = \sqrt{2} + 2$

Step 5.2

Substitute the $x$ value $0$ into $f (x) = \sqrt{- x + 1} + 2$ . In this case, the point is $(0, 3)$ .

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Step 5.2.1

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

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

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- 1$ by $0$ .

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

Step 5.2.2.1.2

Add $0$ and $1$ .

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

Step 5.2.2.1.3

Any root of $1$ is $1$ .

$f (0) = 1 + 2$

Step 5.2.2.2

Add $1$ and $2$ .

$f (0) = 3$

Step 5.2.2.3

The final answer is $3$ .

$y = 3$

Step 5.3

The square root can be graphed using the points around the vertex $(1, 2), (- 1, 3.41), (0, 3)$

$\begin{array}{c} x & y \\ - 1 & 3.41 \\ 0 & 3 \\ 1 & 2 \end{array}$

Step 6