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[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Step Number} & \text{Function} & \text{Step Description} \\
\hline
\text{Step 1} & f(x)=\frac{3 x+4}{8} & \text{given} \\
\hline
\text{Step 2} & y=\frac{3 x+4}{8} & \text{change } f(x) \text{ to } y \\
\hline
\text{Step 3} & x=\frac{3 y+4}{8} & \text{switch } x \text{ and } y \\
\hline
\text{Step 4} & 8 x=3 y+4 & \text{multiply each side by 8} \\
\hline
\text{Step 5} & 8 x-4=3 y & \text{subtract 4 from each side} \\
\hline
\text{Step 6} & 24 x-12=y & \text{multiply each side by 3} \\
\hline
\text{Step 7} & 24 x-12=f^{-1}(x) & \text{replace } y \text{ with } f^{-1}(x) \\
\hline
\end{array}
\][/tex]

Use this information to complete the statement.



Answer :

GinnyAnswer
Certainly! Let’s work through the steps as given in the table to find the inverse function [tex]\( f^{-1}(x) \)[/tex].

### Given
1. Step 1: Function Given
[tex]\[ f(x) = \frac{3x + 4}{8} \][/tex]

2. Step 2: Change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex]
[tex]\[ y = \frac{3x + 4}{8} \][/tex]

3. Step 3: Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
[tex]\[ x = \frac{3y + 4}{8} \][/tex]

### Solving for [tex]\( y \)[/tex]
4. Step 4: Multiply each side by 8 to eliminate the denominator
[tex]\[ 8x = 3y + 4 \][/tex]

5. Step 5: Isolate the term containing [tex]\( y \)[/tex]
[tex]\[ 8x - 4 = 3y \][/tex]

6. Step 6: Solve for [tex]\( y \)[/tex]
[tex]\[ y = \frac{8x - 4}{3} \][/tex]

### Conclusion
7. Step 7: Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex]
[tex]\[ f^{-1}(x) = \frac{8x - 4}{3} \][/tex]

So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{8x - 4}{3} \][/tex]

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