Certainly! Let's go through the process step-by-step to determine the inverse of the given function [tex]\( f(x) = \sqrt{3(8x) + 4} \)[/tex].
1. Write the Original Function with [tex]\( y \)[/tex]:
Given [tex]\( f(x) = \sqrt{3(8x) + 4} \)[/tex], rewrite it as:
[tex]\[ y = \sqrt{3(8x) + 4} \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse function, swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt{3(8y) + 4} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, square both sides of the equation to eliminate the square root:
[tex]\[ x^2 = 3(8y) + 4 \][/tex]
- Isolate the term involving [tex]\( y \)[/tex]:
[tex]\[ x^2 - 4 = 3(8y) \][/tex]
- Divide both sides of the equation by 3 to further isolate the term with [tex]\( y \)[/tex]:
[tex]\[ 8y = \frac{x^2 - 4}{3} \][/tex]
- Finally, divide both sides by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x^2 - 4}{3 \cdot 8} = \frac{x^2 - 4}{24} \][/tex]
4. Write the Inverse Function:
Swap [tex]\( y \)[/tex] back to [tex]\( f^{-1}(x) \)[/tex] to express the inverse function:
[tex]\[ f^{-1}(x) = \frac{x^2 - 4}{24} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x^2 - 4}{24} \][/tex]
This completes the steps to find the inverse of the function [tex]\( f(x) = \sqrt{3(8x) + 4} \)[/tex]. The inverse function [tex]\( f^{-1}(x) \)[/tex] is given by:
[tex]\[ f^{-1}(x) = \frac{x^2 - 4}{24} \][/tex]
