To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 2x + 16 \)[/tex], we will follow these steps:
1. Set [tex]\( y \)[/tex] equal to [tex]\( f(x) \)[/tex]:
[tex]\[ y = 2x + 16 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function. This is because an inverse function reverses the roles of the independent and dependent variables:
[tex]\[ x = 2y + 16 \][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. This involves isolating [tex]\( y \)[/tex]:
a. Subtract 16 from both sides of the equation:
[tex]\[ x - 16 = 2y \][/tex]
b. Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x - 16}{2} \][/tex]
4. Write the inverse function:
Since [tex]\( y \)[/tex] is now expressed in terms of [tex]\( x \)[/tex], we denote this new function as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x - 16}{2} \][/tex]
So, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x - 16}{2} \][/tex]
