Find the inverse for each of the following functions. $$ f(x)=2 x+8 $$ $$ f^{-1}(x)= $$ $$ g(x)=7 x^{3}-7 $$ $$ g^{-1}(x)= $$ To access the cube root, enter the answer cell, click on the yellow arrow at the right and then click on the function tab, $$ h(x)=\frac{2}{x+7} $$ $$ h^{-1}(x)= $$ $$ h^{2} $$ $$ j(x)=\sqrt[3]{x+7} $$ $$ j^{-1}(x)= $$ $$ \square $$ Question Help: $$ \square $$ Message instructor

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$$ \textbf{For } f(x)=2x+8: $$

1. Write $$ y = 2x+8 $$.
2. Solve for $$ x $$:
   $$
   y-8 = 2x \quad \Longrightarrow \quad x = \frac{y-8}{2}
   $$
3. Replace $$ y $$ with $$ x $$ to obtain the inverse:
   $$
   f^{-1}(x)=\frac{x-8}{2}
   $$

$$ \textbf{For } g(x)=7x^{3}-7: $$

1. Write $$ y = 7x^{3}-7 $$.
2. Add 7 to both sides:
   $$
   y+7=7x^{3}
   $$
3. Divide by 7:
   $$
   x^{3}=\frac{y+7}{7}
   $$
4. Take the cube root of both sides:
   $$
   x=\sqrt[3]{\frac{y+7}{7}}
   $$
5. Replace $$ y $$ with $$ x $$ to obtain the inverse:
   $$
   g^{-1}(x)=\sqrt[3]{\frac{x+7}{7}}
   $$

$$ \textbf{For } h(x)=\frac{2}{x+7}: $$

1. Write $$ y=\frac{2}{x+7} $$.
2. Multiply both sides by $$ x+7 $$:
   $$
   y(x+7)=2
   $$
3. Solve for $$ x $$:
   $$
   x+7=\frac{2}{y} \quad \Longrightarrow \quad x=\frac{2}{y}-7
   $$
4. Replace $$ y $$ with $$ x $$ to obtain the inverse:
   $$
   h^{-1}(x)=\frac{2}{x}-7
   $$

$$ \textbf{Compute } h^{2}(x) \text{ (i.e., } h(h(x)) \text{):} $$

1. Substitute $$ h(x)=\frac{2}{x+7} $$ into itself:
   $$
   h(h(x)) = \frac{2}{\left(\frac{2}{x+7}\right)+7}
   $$
2. Combine the terms in the denominator:
   $$
   \frac{2}{x+7}+7=\frac{2+7(x+7)}{x+7}=\frac{7x+49+2}{x+7}=\frac{7x+51}{x+7}
   $$
3. Therefore,
   $$
   h^{2}(x)=\frac{2}{\frac{7x+51}{x+7}}=\frac{2(x+7)}{7x+51}
   $$

$$ \textbf{For } j(x)=\sqrt[3]{x+7}: $$

1. Write $$ y=\sqrt[3]{x+7} $$.
2. Cube both sides:
   $$
   y^{3}=x+7
   $$
3. Solve for $$ x $$:
   $$
   x=y^{3}-7
   $$
4. Replace $$ y $$ with $$ x $$ to obtain the inverse:
   $$
   j^{-1}(x)=x^{3}-7
   $$
Here are the inverses of the given functions: 1. $$ f^{-1}(x) = \frac{x - 8}{2} $$ 2. $$ g^{-1}(x) = \sqrt[3]{\frac{x + 7}{7}} $$ 3. $$ h^{-1}(x) = \frac{2}{x} - 7 $$ 4. $$ h^{2}(x) = \frac{2(x + 7)}{7x + 51} $$ 5. $$ j^{-1}(x) = x^{3} - 7 $$

Find the inverse for each of the following functions.

To access the cube root, enter the answer cell, click on the
yellow arrow at the right and then click on the function tab,

Question Help: Message instructor

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Find the inverse for each of the following functions. To access the cube root, enter the answer cell, click on the yellow arrow at the right and then click on the function tab, Question Help: Message instructor