Find the x and y Intercepts f(x)=1/2( cube root of 24-8x)+1

$f (x) = \frac{1}{2} (\sqrt[3]{24 - 8 x}) + 1$

Find the x-intercepts.

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

$0 = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 x}) + 1$

Solve the equation.

Rewrite the equation as $\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} + 1 = 0$ .

$\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} + 1 = 0$

Subtract $1$ from both sides of the equation.

$\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} = - 1$

To remove the radical on the left side of the equation, cube both sides of the equation.

${(\frac{1}{2} \cdot \sqrt[3]{24 - 8 x})}^{3} = {(- 1)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{24 - 8 x}$ as ${(24 - 8 x)}^{\frac{1}{3}}$ .

${(\frac{1}{2} \cdot {(24 - 8 x)}^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Simplify the left side.

Simplify ${(\frac{1}{2} \cdot {(24 - 8 x)}^{\frac{1}{3}})}^{3}$ .

Combine fractions.

Combine $\frac{1}{2}$ and ${(24 - 8 x)}^{\frac{1}{3}}$ .

${(\frac{{(24 - 8 x)}^{\frac{1}{3}}}{2})}^{3} = {(- 1)}^{3}$

Apply the product rule to $\frac{{(24 - 8 x)}^{\frac{1}{3}}}{2}$ .

$\frac{{({(24 - 8 x)}^{\frac{1}{3}})}^{3}}{2^{3}} = {(- 1)}^{3}$

Simplify the numerator.

Multiply the exponents in ${({(24 - 8 x)}^{\frac{1}{3}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(24 - 8 x)}^{\frac{1}{3} \cdot 3}}{2^{3}} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{{(24 - 8 x)}^{\frac{1}{3} \cdot 3}}{2^{3}} = {(- 1)}^{3}$

Rewrite the expression.

$\frac{{(24 - 8 x)}^{1}}{2^{3}} = {(- 1)}^{3}$

Simplify.

$\frac{24 - 8 x}{2^{3}} = {(- 1)}^{3}$

Factor $8$ out of $24 - 8 x$ .

Factor $8$ out of $24$ .

$\frac{8 (3) - 8 x}{2^{3}} = {(- 1)}^{3}$

Factor $8$ out of $- 8 x$ .

$\frac{8 (3) + 8 (- x)}{2^{3}} = {(- 1)}^{3}$

Factor $8$ out of $8 (3) + 8 (- x)$ .

$\frac{8 (3 - x)}{2^{3}} = {(- 1)}^{3}$

Reduce the expression by cancelling the common factors.

Raise $2$ to the power of $3$ .

$\frac{8 (3 - x)}{8} = {(- 1)}^{3}$

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (3 - x)}{8} = {(- 1)}^{3}$

Divide $3 - x$ by $1$ .

$3 - x = {(- 1)}^{3}$

Simplify the right side.

Raise $- 1$ to the power of $3$ .

$3 - x = - 1$

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- x = - 1 - 3$

Subtract $3$ from $- 1$ .

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

Find the y-intercepts.

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

$y = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 \cdot 0}) + 1$

Solve the equation.

Multiply $\frac{1}{2}$ by $\sqrt[3]{24 - 8 \cdot 0}$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24 - 8 \cdot 0} + 1$

Remove parentheses.

$y = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 \cdot 0}) + 1$

Simplify each term.

Multiply $- 8$ by $0$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24 + 0} + 1$

Add $24$ and $0$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24} + 1$

Rewrite $24$ as $2^{3} \cdot 3$ .

Factor $8$ out of $24$ .

$y = \frac{1}{2} \cdot \sqrt[3]{8 (3)} + 1$

Rewrite $8$ as $2^{3}$ .

$y = \frac{1}{2} \cdot \sqrt[3]{2^{3} \cdot 3} + 1$

Pull terms out from under the radical.

$y = \frac{1}{2} \cdot (2 \sqrt[3]{3}) + 1$

Cancel the common factor of $2$ .

Factor $2$ out of $2 \sqrt[3]{3}$ .

$y = \frac{1}{2} \cdot (2 (\sqrt[3]{3})) + 1$

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \sqrt[3]{3}) + 1$

Rewrite the expression.

$y = \sqrt[3]{3} + 1$

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt[3]{3} + 1)$

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, \sqrt[3]{3} + 1)$

Find the x and y Intercepts f(x)=1/2( cube root of 24-8x)+1

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