Pre-Algebra Examples

$f (x) = \sqrt{7 x - 3}$

Step 1

Find the domain for $y = \sqrt{7 x - 3}$ 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 $\sqrt{7 x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$7 x - 3 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $3$ to both sides of the inequality.

$7 x \geq 3$

Step 1.2.2

Divide each term in $7 x \geq 3$ by $7$ and simplify.

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Step 1.2.2.1

Divide each term in $7 x \geq 3$ by $7$ .

$\frac{7 x}{7} \geq \frac{3}{7}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{3}{7}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{3}{7}$

Step 1.3

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

Interval Notation:

$[\frac{3}{7}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{3}{7}}$

Interval Notation:

$[\frac{3}{7}, \infty)$

Set-Builder Notation:

${x | x \geq \frac{3}{7}}$

Step 2

To find the radical expression end point, substitute the $x$ value $\frac{3}{7}$ , which is the least value in the domain, into $f (x) = \sqrt{7 x - 3}$ .

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Step 2.1

Replace the variable $x$ with $\frac{3}{7}$ in the expression.

$f (\frac{3}{7}) = \sqrt{7 (\frac{3}{7}) - 3}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$f (\frac{3}{7}) = \sqrt{7 (\frac{3}{7}) - 3}$

Step 2.2.1.2

Rewrite the expression.

$f (\frac{3}{7}) = \sqrt{3 - 3}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Subtract $3$ from $3$ .

$f (\frac{3}{7}) = \sqrt{0}$

Step 2.2.2.2

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

$f (\frac{3}{7}) = \sqrt{0^{2}}$

Step 2.2.2.3

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

$f (\frac{3}{7}) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(\frac{3}{7}, 0)$ .

$(\frac{3}{7}, 0)$

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 $1$ into $f (x) = \sqrt{7 x - 3}$ . In this case, the point is $(1, 2)$ .

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Step 4.1.1

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

$f (1) = \sqrt{7 (1) - 3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $7$ by $1$ .

$f (1) = \sqrt{7 - 3}$

Step 4.1.2.2

Subtract $3$ from $7$ .

$f (1) = \sqrt{4}$

Step 4.1.2.3

Rewrite $4$ as $2^{2}$ .

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

Step 4.1.2.4

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

$f (1) = 2$

Step 4.1.2.5

The final answer is $2$ .

$y = 2$

Step 4.2

Substitute the $x$ value $2$ into $f (x) = \sqrt{7 x - 3}$ . In this case, the point is $(2, \sqrt{11})$ .

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Step 4.2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \sqrt{7 (2) - 3}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $7$ by $2$ .

$f (2) = \sqrt{14 - 3}$

Step 4.2.2.2

Subtract $3$ from $14$ .

$f (2) = \sqrt{11}$

Step 4.2.2.3

The final answer is $\sqrt{11}$ .

$y = \sqrt{11}$

Step 4.3

The square root can be graphed using the points around the vertex $(0.\overline{428571}, 0), (1, 2), (2, 3.32)$

$\begin{array}{c} x & y \\ 0.429 & 0 \\ 1 & 2 \\ 2 & 3.32 \end{array}$

Step 5