Pre-Algebra Examples

$\sqrt{4 x + 1}$

Step 1

Find the domain for $y = \sqrt{4 x + 1}$ 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 $\sqrt{4 x + 1}$ greater than or equal to $0$ to find where the expression is defined.

$4 x + 1 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $1$ from both sides of the inequality.

$4 x \geq - 1$

Step 1.2.2

Divide each term in $4 x \geq - 1$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 x \geq - 1$ by $4$ .

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 1}{4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{1}{4}$

Step 1.3

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

Interval Notation:

$[- \frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{1}{4}}$

Interval Notation:

$[- \frac{1}{4}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{1}{4}}$

Step 2

To find the radical expression end point, substitute the $x$ value $- \frac{1}{4}$ , which is the least value in the domain, into $f (x) = \sqrt{4 x + 1}$ .

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Step 2.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

$f (- \frac{1}{4}) = \sqrt{4 (- \frac{1}{4}) + 1}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (- \frac{1}{4}) = \sqrt{4 (\frac{- 1}{4}) + 1}$

Step 2.2.1.2

Cancel the common factor.

$f (- \frac{1}{4}) = \sqrt{4 (\frac{- 1}{4}) + 1}$

Step 2.2.1.3

Rewrite the expression.

$f (- \frac{1}{4}) = \sqrt{- 1 + 1}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Add $- 1$ and $1$ .

$f (- \frac{1}{4}) = \sqrt{0}$

Step 2.2.2.2

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

$f (- \frac{1}{4}) = \sqrt{0^{2}}$

Step 2.2.2.3

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

$f (- \frac{1}{4}) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(- \frac{1}{4}, 0)$ .

$(- \frac{1}{4}, 0)$

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 $0$ into $f (x) = \sqrt{4 x + 1}$ . In this case, the point is $(- 0, 1)$ .

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Step 4.1.1

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

$f (0) = \sqrt{4 (0) + 1}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $4$ by $0$ .

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

Step 4.1.2.2

Add $0$ and $1$ .

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

Step 4.1.2.3

Any root of $1$ is $1$ .

$f (0) = 1$

Step 4.1.2.4

The final answer is $1$ .

$y = 1$

Step 4.2

Substitute the $x$ value $1$ into $f (x) = \sqrt{4 x + 1}$ . In this case, the point is $(1, \sqrt{5})$ .

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $4$ by $1$ .

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

Step 4.2.2.2

Add $4$ and $1$ .

$f (1) = \sqrt{5}$

Step 4.2.2.3

The final answer is $\sqrt{5}$ .

$y = \sqrt{5}$

Step 4.3

Substitute the $x$ value $2$ into $f (x) = \sqrt{4 x + 1}$ . In this case, the point is $(2, 3)$ .

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Step 4.3.1

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

$f (2) = \sqrt{4 (2) + 1}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Multiply $4$ by $2$ .

$f (2) = \sqrt{8 + 1}$

Step 4.3.2.2

Add $8$ and $1$ .

$f (2) = \sqrt{9}$

Step 4.3.2.3

Rewrite $9$ as $3^{2}$ .

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

Step 4.3.2.4

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

$f (2) = 3$

Step 4.3.2.5

The final answer is $3$ .

$y = 3$

Step 4.4

The square root can be graphed using the points around the vertex $(- 0.25, 0), (- 0, 1), (1, 2.24), (2, 3)$

$\begin{array}{c} x & y \\ - 0.25 & 0 \\ - 0 & 1 \\ 1 & 2.24 \\ 2 & 3 \end{array}$

Step 5