Find the Domain and Range F(x)=(2x^2-5)/(x^2)

$F (x) = \frac{2 x^{2} - 5}{x^{2}}$

Step 1

Set the denominator in $\frac{2 x^{2} - 5}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 2

Solve for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$

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

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

Step 6