Finite Math Examples

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

Step 1

Set the radicand in $\sqrt{6 - 7 x}$ greater than or equal to $0$ to find where the expression is defined.

$6 - 7 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$- 7 x \geq - 6$

Step 2.2

Divide each term in $- 7 x \geq - 6$ by $- 7$ and simplify.

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Step 2.2.1

Divide each term in $- 7 x \geq - 6$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 6}{- 7}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 6}{- 7}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 7}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{6}{7}$

Step 3

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

Interval Notation:

$(- \infty, \frac{6}{7}]$

Set-Builder Notation:

${x | x \leq \frac{6}{7}}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{6}{7}], {x | x \leq \frac{6}{7}}$

Range: $[0, \infty), {y | y \geq 0}$

Step 6