Find the Domain and Range square root of x^2-7x+10

$\sqrt{x^{2} - 7 x + 10}$

Step 1

Set the radicand in $\sqrt{x^{2} - 7 x + 10}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 7 x + 10 \geq 0$

Step 2

Solve for $x$ .

Convert the inequality to an equation.

$x^{2} - 7 x + 10 = 0$

Factor $x^{2} - 7 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$(x - 5) (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x - 2 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 5) (x - 2) = 0$ true.

$x = 5, 2$

Use each root to create test intervals.

$x < 2$

$2 < x < 5$

$x > 5$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 2$ to see if it makes the inequality true.

Choose a value on the interval $x < 2$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

${(0)}^{2} - 7 \cdot 0 + 10 \geq 0$

The left side $10$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

${(4)}^{2} - 7 \cdot 4 + 10 \geq 0$

The left side $- 2$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

${(8)}^{2} - 7 \cdot 8 + 10 \geq 0$

The left side $18$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

The solution consists of all of the true intervals.

$x \leq 2$ or $x \geq 5$

Step 3

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

Interval Notation:

$(- \infty, 2] \cup [5, \infty)$

Set-Builder Notation:

${x | x \leq 2, x \geq 5}$

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: $(- \infty, 2] \cup [5, \infty), {x | x \leq 2, x \geq 5}$

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

Step 6

Find the Domain and Range square root of x^2-7x+10

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