Find the Domain y=(12-3x)/(x^2-7x+12)

$y = \frac{12 - 3 x}{x^{2} - 7 x + 12}$

Step 1

Set the denominator in $\frac{12 - 3 x}{x^{2} - 7 x + 12}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 7 x + 12 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

$(x - 4) (x - 3) = 0$

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

$x - 4 = 0$

$x - 3 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x - 4) (x - 3) = 0$ true.

$x = 4, 3$

Step 3

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

Interval Notation:

$(- \infty, 3) \cup (3, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 3, 4}$

Step 4

Find the Domain y=(12-3x)/(x^2-7x+12)

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