Algebra Examples

y = \sqrt{x}

$y = \sqrt{x}$

Step 1

Find the domain for

y = \sqrt{x}

$y = \sqrt{x}$ so that a list of

x

$x$ 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{x}

$\sqrt{x}$ greater than or equal to

0

$0$ to find where the expression is defined.

x \geq 0

$x \geq 0$

Step 1.2

The domain is all values of

x

$x$ that make the expression defined.

Interval Notation:

[0, \infty)

$[0, \infty)$

Set-Builder Notation:

{x | x \geq 0}

${x | x \geq 0}$

Interval Notation:

[0, \infty)

$[0, \infty)$

Set-Builder Notation:

{x | x \geq 0}

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the

x

$x$ value

0

$0$ , which is the least value in the domain, into

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ .

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Step 2.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = \sqrt{0}

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

f (0) = \sqrt{0}

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

Step 2.2.2

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

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

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

Step 2.2.3

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

f (0) = 0

$f (0) = 0$

Step 2.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

0

$0$

Step 3

The radical expression end point is

(0, 0)

$(0, 0)$ .

(0, 0)

$(0, 0)$

Step 4

Select a few

x

$x$ values from the domain. It would be more useful to select the values so that they are next to the

x

$x$ value of the radical expression end point.

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Step 4.1

Substitute the

x

$x$ value

1

$1$ into

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ . In this case, the point is

(1, 1)

$(1, 1)$ .

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Step 4.1.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = \sqrt{1}

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

f (1) = \sqrt{1}

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

Step 4.1.2.2

Any root of

1

$1$ is

1

$1$ .

f (1) = 1

$f (1) = 1$

Step 4.1.2.3

The final answer is

1

$1$ .

y = 1

$y = 1$

y = 1

$y = 1$

y = 1

$y = 1$

Step 4.2

Substitute the

x

$x$ value

2

$2$ into

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ . In this case, the point is

(2, \sqrt{2})

$(2, \sqrt{2})$ .

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Step 4.2.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = \sqrt{2}

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Remove parentheses.

f (2) = \sqrt{2}

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

Step 4.2.2.2

The final answer is

\sqrt{2}

$\sqrt{2}$ .

y = \sqrt{2}

$y = \sqrt{2}$

y = \sqrt{2}

$y = \sqrt{2}$

y = \sqrt{2}

$y = \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex

(0, 0), (1, 1), (2, 1.41)

$(0, 0), (1, 1), (2, 1.41)$

\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.41 \end{array}

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.41 \end{array}$

\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.41 \end{array}

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 1 \\ 2 & 1.41 \end{array}$

Step 5