Find g(f(x)) f(x) = cube root of (x-3)/7 , g(x)=7x^3+3

$f (x) = \sqrt[3]{\frac{x - 3}{7}}$ , $g (x) = 7 x^{3} + 3$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\sqrt[3]{\frac{x - 3}{7}})$ by substituting in the value of $f$ into $g$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\sqrt[3]{\frac{x - 3}{7}})}^{3} + 3$

Step 3

Simplify each term.

Rewrite $\sqrt[3]{\frac{x - 3}{7}}$ as $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{7}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3}}{\sqrt[3]{7}})}^{3} + 3$

Multiply $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}})}^{3} + 3$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{x - 3}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}})}^{3} + 3$

Raise $\sqrt[3]{7}$ to the power of $1$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}})}^{3} + 3$

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

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}})}^{3} + 3$

Add $1$ and $2$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}})}^{3} + 3$

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}})}^{3} + 3$

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

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}})}^{3} + 3$

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

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}})}^{3} + 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}})}^{3} + 3$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{7})}^{3} + 3$

Evaluate the exponent.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} {\sqrt[3]{7}}^{2}}{7})}^{3} + 3$

Simplify the numerator.

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} \sqrt[3]{7^{2}}}{7})}^{3} + 3$

Raise $7$ to the power of $2$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{x - 3} \sqrt[3]{49}}{7})}^{3} + 3$

Combine using the product rule for radicals.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 {(\frac{\sqrt[3]{(x - 3) \cdot 49}}{7})}^{3} + 3$

Apply the product rule to $\frac{\sqrt[3]{(x - 3) \cdot 49}}{7}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{{\sqrt[3]{(x - 3) \cdot 49}}^{3}}{7^{3}}) + 3$

Simplify the numerator.

Rewrite ${\sqrt[3]{(x - 3) \cdot 49}}^{3}$ as $(x - 3) \cdot 49$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{(x - 3) \cdot 49}$ as ${((x - 3) \cdot 49)}^{\frac{1}{3}}$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{{({((x - 3) \cdot 49)}^{\frac{1}{3}})}^{3}}{7^{3}}) + 3$

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

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{{((x - 3) \cdot 49)}^{\frac{1}{3} \cdot 3}}{7^{3}}) + 3$

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

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{{((x - 3) \cdot 49)}^{\frac{3}{3}}}{7^{3}}) + 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{{((x - 3) \cdot 49)}^{\frac{3}{3}}}{7^{3}}) + 3$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{(x - 3) \cdot 49}{7^{3}}) + 3$

Simplify.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{(x - 3) \cdot 49}{7^{3}}) + 3$

Apply the distributive property.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{x \cdot 49 - 3 \cdot 49}{7^{3}}) + 3$

Move $49$ to the left of $x$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 \cdot x - 3 \cdot 49}{7^{3}}) + 3$

Multiply $- 3$ by $49$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 \cdot x - 147}{7^{3}}) + 3$

Factor $49$ out of $49 x - 147$ .

Factor $49$ out of $49 x$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 (x) - 147}{7^{3}}) + 3$

Factor $49$ out of $- 147$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 x + 49 \cdot - 3}{7^{3}}) + 3$

Factor $49$ out of $49 x + 49 \cdot - 3$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 (x - 3)}{7^{3}}) + 3$

Raise $7$ to the power of $3$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 (x - 3)}{343}) + 3$

Cancel the common factor of $7$ .

Factor $7$ out of $343$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 (x - 3)}{7 (49)}) + 3$

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 3}{7}}) = 7 (\frac{49 (x - 3)}{7 \cdot 49}) + 3$

Rewrite the expression.

$g (\sqrt[3]{\frac{x - 3}{7}}) = \frac{49 (x - 3)}{49} + 3$

Cancel the common factor of $49$ .

Cancel the common factor.

$g (\sqrt[3]{\frac{x - 3}{7}}) = \frac{49 (x - 3)}{49} + 3$

Divide $x - 3$ by $1$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = x - 3 + 3$

Step 4

Combine the opposite terms in $x - 3 + 3$ .

Add $- 3$ and $3$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = x + 0$

Add $x$ and $0$ .

$g (\sqrt[3]{\frac{x - 3}{7}}) = x$

Find g(f(x)) f(x) = cube root of (x-3)/7 , g(x)=7x^3+3

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