Find the Domain and Range f(x)=-1/((x+3)^2)+2

$f (x) = - \frac{1}{{(x + 3)}^{2}} + 2$

Step 1

Set the denominator in $\frac{1}{{(x + 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq - 3}$

Step 4

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

Interval Notation:

$(- \infty, 2)$

Set-Builder Notation:

${y | y < 2}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, 2), {y | y < 2}$

Step 6