Algebra Examples

$f (x) = \frac{3}{x + 2} - \sqrt{x - 3}$

Step 1

Find the domain for $y = \frac{3}{x + 2} - \sqrt{x - 3}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt{x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$x - 3 \geq 0$

Step 1.2

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 1.3

Set the denominator in $\frac{3 - \sqrt{x - 3} x - 2 \sqrt{x - 3}}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 1.4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.5

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

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${x | x \geq 3}$

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${x | x \geq 3}$

Step 2

To find the radical expression end point, substitute the $x$ value $3$ , which is the least value in the domain, into $f (x) = \frac{3 - \sqrt{x - 3} x - 2 \sqrt{x - 3}}{x + 2}$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{3 - \sqrt{(3) - 3} \cdot 3 - 2 \sqrt{(3) - 3}}{(3) + 2}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.1

Subtract $3$ from $3$ .

$f (3) = \frac{3 - \sqrt{0} \cdot 3 - 2 \sqrt{3 - 3}}{3 + 2}$

Step 2.2.1.2

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

$f (3) = \frac{3 - \sqrt{0^{2}} \cdot 3 - 2 \sqrt{3 - 3}}{3 + 2}$

Step 2.2.1.3

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

$f (3) = \frac{3 + 0 \cdot 3 - 2 \sqrt{3 - 3}}{3 + 2}$

Step 2.2.1.4

Multiply $- 0 \cdot 3$ .

Tap for more steps...

Step 2.2.1.4.1

Multiply $- 1$ by $0$ .

$f (3) = \frac{3 + 0 \cdot 3 - 2 \sqrt{3 - 3}}{3 + 2}$

Step 2.2.1.4.2

Multiply $0$ by $3$ .

$f (3) = \frac{3 + 0 - 2 \sqrt{3 - 3}}{3 + 2}$

Step 2.2.1.5

Subtract $3$ from $3$ .

$f (3) = \frac{3 + 0 - 2 \sqrt{0}}{3 + 2}$

Step 2.2.1.6

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

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

Step 2.2.1.7

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

$f (3) = \frac{3 + 0 - 2 \cdot 0}{3 + 2}$

Step 2.2.1.8

Multiply $- 2$ by $0$ .

$f (3) = \frac{3 + 0 + 0}{3 + 2}$

Step 2.2.1.9

Add $3 + 0$ and $0$ .

$f (3) = \frac{3 + 0}{3 + 2}$

Step 2.2.1.10

Add $3$ and $0$ .

$f (3) = \frac{3}{3 + 2}$

Step 2.2.2

Add $3$ and $2$ .

$f (3) = \frac{3}{5}$

Step 2.2.3

The final answer is $\frac{3}{5}$ .

$\frac{3}{5}$

Step 3

The radical expression end point is $(3, \frac{3}{5})$ .

$(3, \frac{3}{5})$

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.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{3 - \sqrt{(4) - 3} \cdot 4 - 2 \sqrt{(4) - 3}}{(4) + 2}$

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

Simplify the numerator.

Tap for more steps...

Step 4.1.2.1.1

Subtract $3$ from $4$ .

$f (4) = \frac{3 - \sqrt{1} \cdot 4 - 2 \sqrt{4 - 3}}{4 + 2}$

Step 4.1.2.1.2

Any root of $1$ is $1$ .

$f (4) = \frac{3 - 1 \cdot 1 \cdot 4 - 2 \sqrt{4 - 3}}{4 + 2}$

Step 4.1.2.1.3

Multiply $- 1 \cdot 1 \cdot 4$ .

Tap for more steps...

Step 4.1.2.1.3.1

Multiply $- 1$ by $1$ .

$f (4) = \frac{3 - 1 \cdot 4 - 2 \sqrt{4 - 3}}{4 + 2}$

Step 4.1.2.1.3.2

Multiply $- 1$ by $4$ .

$f (4) = \frac{3 - 4 - 2 \sqrt{4 - 3}}{4 + 2}$

Step 4.1.2.1.4

Subtract $3$ from $4$ .

$f (4) = \frac{3 - 4 - 2 \sqrt{1}}{4 + 2}$

Step 4.1.2.1.5

Any root of $1$ is $1$ .

$f (4) = \frac{3 - 4 - 2 \cdot 1}{4 + 2}$

Step 4.1.2.1.6

Multiply $- 2$ by $1$ .

$f (4) = \frac{3 - 4 - 2}{4 + 2}$

Step 4.1.2.1.7

Subtract $4$ from $3$ .

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

Step 4.1.2.1.8

Subtract $2$ from $- 1$ .

$f (4) = \frac{- 3}{4 + 2}$

Step 4.1.2.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.1.2.2.1

Add $4$ and $2$ .

$f (4) = \frac{- 3}{6}$

Step 4.1.2.2.2

Cancel the common factor of $- 3$ and $6$ .

Tap for more steps...

Step 4.1.2.2.2.1

Factor $3$ out of $- 3$ .

$f (4) = \frac{3 (- 1)}{6}$

Step 4.1.2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.2.2.2.1

Factor $3$ out of $6$ .

$f (4) = \frac{3 \cdot - 1}{3 \cdot 2}$

Step 4.1.2.2.2.2.2

Cancel the common factor.

$f (4) = \frac{3 \cdot - 1}{3 \cdot 2}$

Step 4.1.2.2.2.2.3

Rewrite the expression.

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

Step 4.1.2.2.3

Move the negative in front of the fraction.

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

Step 4.1.2.3

The final answer is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 4.2

Substitute the $x$ value $5$ into $f (x) = \frac{3 - \sqrt{x - 3} x - 2 \sqrt{x - 3}}{x + 2}$ . In this case, the point is $(5, \frac{3 - 7 \sqrt{2}}{7})$ .

Tap for more steps...

Step 4.2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{3 - \sqrt{(5) - 3} \cdot 5 - 2 \sqrt{(5) - 3}}{(5) + 2}$

Step 4.2.2

Simplify the result.

Tap for more steps...

Step 4.2.2.1

Simplify the numerator.

Tap for more steps...

Step 4.2.2.1.1

Subtract $3$ from $5$ .

$f (5) = \frac{3 - \sqrt{2} \cdot 5 - 2 \sqrt{5 - 3}}{5 + 2}$

Step 4.2.2.1.2

Multiply $5$ by $- 1$ .

$f (5) = \frac{3 - 5 \sqrt{2} - 2 \sqrt{5 - 3}}{5 + 2}$

Step 4.2.2.1.3

Subtract $3$ from $5$ .

$f (5) = \frac{3 - 5 \sqrt{2} - 2 \sqrt{2}}{5 + 2}$

Step 4.2.2.1.4

Subtract $2 \sqrt{2}$ from $- 5 \sqrt{2}$ .

$f (5) = \frac{3 - 7 \sqrt{2}}{5 + 2}$

Step 4.2.2.2

Add $5$ and $2$ .

$f (5) = \frac{3 - 7 \sqrt{2}}{7}$

Step 4.2.2.3

The final answer is $\frac{3 - 7 \sqrt{2}}{7}$ .

$y = \frac{3 - 7 \sqrt{2}}{7}$

Step 4.3

The square root can be graphed using the points around the vertex $(3, \frac{3}{5}), (4, - 0.5), (5, - 0.99)$

$\begin{array}{c} x & y \\ 3 & \frac{3}{5} \\ 4 & - 0.5 \\ 5 & - 0.99 \end{array}$

Step 5