Find g(f(x)) f(x) = cube root of (x-2)/6 , g(x)=6x^3+2

$f (x) = \sqrt[3]{\frac{x - 2}{6}}$ , $g (x) = 6 x^{3} + 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\sqrt[3]{\frac{x - 2}{6}})$ by substituting in the value of $f$ into $g$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\sqrt[3]{\frac{x - 2}{6}})}^{3} + 2$

Step 3

Simplify each term.

Rewrite $\sqrt[3]{\frac{x - 2}{6}}$ as $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{6}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2}}{\sqrt[3]{6}})}^{3} + 2$

Multiply $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2}}{\sqrt[3]{6}} \cdot \frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}})}^{3} + 2$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{x - 2}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}})}^{3} + 2$

Raise $\sqrt[3]{6}$ to the power of $1$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}})}^{3} + 2$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1 + 2}})}^{3} + 2$

Add $1$ and $2$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{3}})}^{3} + 2$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{{(6^{\frac{1}{3}})}^{3}})}^{3} + 2$

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

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{6^{\frac{1}{3} \cdot 3}})}^{3} + 2$

Combine $\frac{1}{3}$ and $3$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})}^{3} + 2$

Cancel the common factor of $3$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})}^{3} + 2$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{6})}^{3} + 2$

Evaluate the exponent.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} {\sqrt[3]{6}}^{2}}{6})}^{3} + 2$

Simplify the numerator.

Rewrite ${\sqrt[3]{6}}^{2}$ as $\sqrt[3]{6^{2}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} \sqrt[3]{6^{2}}}{6})}^{3} + 2$

Raise $6$ to the power of $2$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{x - 2} \sqrt[3]{36}}{6})}^{3} + 2$

Combine using the product rule for radicals.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 {(\frac{\sqrt[3]{(x - 2) \cdot 36}}{6})}^{3} + 2$

Apply the product rule to $\frac{\sqrt[3]{(x - 2) \cdot 36}}{6}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{{\sqrt[3]{(x - 2) \cdot 36}}^{3}}{6^{3}}) + 2$

Simplify the numerator.

Rewrite ${\sqrt[3]{(x - 2) \cdot 36}}^{3}$ as $(x - 2) \cdot 36$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{(x - 2) \cdot 36}$ as ${((x - 2) \cdot 36)}^{\frac{1}{3}}$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{{({((x - 2) \cdot 36)}^{\frac{1}{3}})}^{3}}{6^{3}}) + 2$

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

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{{((x - 2) \cdot 36)}^{\frac{1}{3} \cdot 3}}{6^{3}}) + 2$

Combine $\frac{1}{3}$ and $3$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{{((x - 2) \cdot 36)}^{\frac{3}{3}}}{6^{3}}) + 2$

Cancel the common factor of $3$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{{((x - 2) \cdot 36)}^{\frac{3}{3}}}{6^{3}}) + 2$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{(x - 2) \cdot 36}{6^{3}}) + 2$

Simplify.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{(x - 2) \cdot 36}{6^{3}}) + 2$

Apply the distributive property.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{x \cdot 36 - 2 \cdot 36}{6^{3}}) + 2$

Move $36$ to the left of $x$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 \cdot x - 2 \cdot 36}{6^{3}}) + 2$

Multiply $- 2$ by $36$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 \cdot x - 72}{6^{3}}) + 2$

Factor $36$ out of $36 x - 72$ .

Factor $36$ out of $36 x$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 (x) - 72}{6^{3}}) + 2$

Factor $36$ out of $- 72$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 x + 36 \cdot - 2}{6^{3}}) + 2$

Factor $36$ out of $36 x + 36 \cdot - 2$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 (x - 2)}{6^{3}}) + 2$

Raise $6$ to the power of $3$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 (x - 2)}{216}) + 2$

Cancel the common factor of $6$ .

Factor $6$ out of $216$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 (x - 2)}{6 (36)}) + 2$

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 2}{6}}) = 6 (\frac{36 (x - 2)}{6 \cdot 36}) + 2$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 2}{6}}) = \frac{36 (x - 2)}{36} + 2$

Cancel the common factor of $36$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 2}{6}}) = \frac{36 (x - 2)}{36} + 2$

Divide $x - 2$ by $1$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = x - 2 + 2$

Step 4

Combine the opposite terms in $x - 2 + 2$ .

Add $- 2$ and $2$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = x + 0$

Add $x$ and $0$ .

$g (\sqrt[3]{\frac{x - 2}{6}}) = x$

Find g(f(x)) f(x) = cube root of (x-2)/6 , g(x)=6x^3+2

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