Find the x and y Intercepts f(x)=(x^2-9)/(2x^2+1)

$f (x) = \frac{x^{2} - 9}{2 x^{2} + 1}$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{x^{2} - 9}{2 x^{2} + 1}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} - 9 = 0$

Solve the equation for $x$ .

Add $9$ to both sides of the equation.

$x^{2} = 9$

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

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

$x = \pm 3$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{{(0)}^{2} - 9}{2 {(0)}^{2} + 1}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} - 9}{2 {(0)}^{2} + 1}$

Remove parentheses.

$y = \frac{0^{2} - 9}{2 \cdot 0^{2} + 1}$

Remove parentheses.

$y = \frac{{(0)}^{2} - 9}{2 {(0)}^{2} + 1}$

Simplify $\frac{{(0)}^{2} - 9}{2 {(0)}^{2} + 1}$ .

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 - 9}{2 \cdot 0^{2} + 1}$

Subtract $9$ from $0$ .

$y = \frac{- 9}{2 \cdot 0^{2} + 1}$

Simplify the denominator.

Raising $0$ to any positive power yields $0$ .

$y = \frac{- 9}{2 \cdot 0 + 1}$

Multiply $2$ by $0$ .

$y = \frac{- 9}{0 + 1}$

Add $0$ and $1$ .

$y = \frac{- 9}{1}$

Divide $- 9$ by $1$ .

$y = - 9$

y-intercept(s) in point form.

y-intercept(s): $(0, - 9)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 3, 0)$

y-intercept(s): $(0, - 9)$

Step 4

Find the x and y Intercepts f(x)=(x^2-9)/(2x^2+1)

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