Find the x and y Intercepts h(x)=(x^3+6)/2

$h (x) = \frac{x^{3} + 6}{2}$

Step 1

Find the x-intercepts.

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

$0 = \frac{x^{3} + 6}{2}$

Solve the equation.

Set the numerator equal to zero.

$x^{3} + 6 = 0$

Solve the equation for $x$ .

Subtract $6$ from both sides of the equation.

$x^{3} = - 6$

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

$x = \sqrt[3]{- 6}$

Simplify $\sqrt[3]{- 6}$ .

Rewrite $- 6$ as ${(- 1)}^{3} \cdot 6$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x = \sqrt[3]{- 1 (6)}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 6}$

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{6}$

Rewrite $- 1 \sqrt[3]{6}$ as $- \sqrt[3]{6}$ .

$x = - \sqrt[3]{6}$

x-intercept(s) in point form.

x-intercept(s): $(- \sqrt[3]{6}, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{{(0)}^{3} + 6}{2}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{3} + 6}{2}$

Remove parentheses.

$y = \frac{{(0)}^{3} + 6}{2}$

Simplify $\frac{{(0)}^{3} + 6}{2}$ .

Simplify the numerator.

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

$y = \frac{0 + 6}{2}$

Add $0$ and $6$ .

$y = \frac{6}{2}$

Divide $6$ by $2$ .

$y = 3$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(- \sqrt[3]{6}, 0)$

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

Step 4

Find the x and y Intercepts h(x)=(x^3+6)/2

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