Answer :

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]