Calculus Examples

$y = \sqrt{7 - x}$

Step 1

Find the domain for $y = \sqrt{7 - x}$ 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 1.1

Set the radicand in $\sqrt{7 - x}$ greater than or equal to $0$ to find where the expression is defined.

$7 - x \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $7$ from both sides of the inequality.

$- x \geq - 7$

Step 1.2.2

Divide each term in $- x \geq - 7$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 7}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 7}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 7$ by $- 1$ .

$x \leq 7$

Step 1.3

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

Interval Notation:

$(- \infty, 7]$

Set-Builder Notation:

${x | x \leq 7}$

Interval Notation:

$(- \infty, 7]$

Set-Builder Notation:

${x | x \leq 7}$

Step 2

To find the radical expression end point, substitute the $x$ value $7$ , which is the least value in the domain, into $f (x) = \sqrt{7 - x}$ .

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Step 2.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \sqrt{7 - (7)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $- 1$ by $7$ .

$f (7) = \sqrt{7 - 7}$

Step 2.2.2

Subtract $7$ from $7$ .

$f (7) = \sqrt{0}$

Step 2.2.3

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

$f (7) = \sqrt{0^{2}}$

Step 2.2.4

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

$f (7) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(7, 0)$ .

$(7, 0)$

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.

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Step 4.1

Substitute the $x$ value $5$ into $f (x) = \sqrt{7 - x}$ . In this case, the point is $(5, \sqrt{2})$ .

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Step 4.1.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \sqrt{7 - (5)}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 1$ by $5$ .

$f (5) = \sqrt{7 - 5}$

Step 4.1.2.2

Subtract $5$ from $7$ .

$f (5) = \sqrt{2}$

Step 4.1.2.3

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

$y = \sqrt{2}$

Step 4.2

Substitute the $x$ value $6$ into $f (x) = \sqrt{7 - x}$ . In this case, the point is $(6, 1)$ .

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Step 4.2.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \sqrt{7 - (6)}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $- 1$ by $6$ .

$f (6) = \sqrt{7 - 6}$

Step 4.2.2.2

Subtract $6$ from $7$ .

$f (6) = \sqrt{1}$

Step 4.2.2.3

Any root of $1$ is $1$ .

$f (6) = 1$

Step 4.2.2.4

The final answer is $1$ .

$y = 1$

Step 4.3

The square root can be graphed using the points around the vertex $(7, 0), (5, 1.41), (6, 1)$

$\begin{array}{c} x & y \\ 5 & 1.41 \\ 6 & 1 \\ 7 & 0 \end{array}$

Step 5