Find the Domain and Range f(x)=(4x)/(x^3+2x^2-3x)

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

Step 1

Set the denominator in $\frac{4 x}{x^{3} + 2 x^{2} - 3 x}$ equal to $0$ to find where the expression is undefined.

$x^{3} + 2 x^{2} - 3 x = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x$ out of $x^{3} + 2 x^{2} - 3 x$ .

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} + 2 x^{2} - 3 x = 0$

Factor $x$ out of $2 x^{2}$ .

$x \cdot x^{2} + x (2 x) - 3 x = 0$

Factor $x$ out of $- 3 x$ .

$x \cdot x^{2} + x (2 x) + x \cdot - 3 = 0$

Factor $x$ out of $x \cdot x^{2} + x (2 x)$ .

$x (x^{2} + 2 x) + x \cdot - 3 = 0$

Factor $x$ out of $x (x^{2} + 2 x) + x \cdot - 3$ .

$x (x^{2} + 2 x - 3) = 0$

Factor.

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

$x ((x - 1) (x + 3)) = 0$

Remove unnecessary parentheses.

$x (x - 1) (x + 3) = 0$

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

$x = 0$

$x - 1 = 0$

$x + 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $x (x - 1) (x + 3) = 0$ true.

$x = 0, 1, - 3$

Step 3

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, 0) \cup (0, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 0, 1, - 3}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - 1] \cup (0, \infty)$

Set-Builder Notation:

${y | y \leq - 1, y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 3) \cup (- 3, 0) \cup (0, 1) \cup (1, \infty), {x | x \neq 0, 1, - 3}$

Range: $(- \infty, - 1] \cup (0, \infty), {y | y \leq - 1, y > 0}$

Step 6

Find the Domain and Range f(x)=(4x)/(x^3+2x^2-3x)

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