Calculus Examples

$f (x) = x^{2} + \sqrt{7 - x}$

Step 1

Find the domain to determine if the expression is continuous.

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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

The expression is continuous.

Continuous

Step 3