Given that [tex]\( f(18) = 2 \)[/tex] and the function [tex]\( f \)[/tex] is one-to-one, we need to find the value of [tex]\( f^{-1}(2) \)[/tex].
Here is a step-by-step explanation:
1. Understand the Function: The function [tex]\( f \)[/tex] is one-to-one, also known as injective, which means each output is associated with exactly one input. This property ensures that every element in the codomain is mapped from a unique element in the domain.
2. Given Information: We are provided with [tex]\( f(18) = 2 \)[/tex]. This signifies that when 18 is input into the function [tex]\( f \)[/tex], the result is 2.
3. Inverse Function Concept: The inverse function [tex]\( f^{-1} \)[/tex] reverses the mapping of [tex]\( f \)[/tex]. Specifically, if [tex]\( f(a) = b \)[/tex], then [tex]\( f^{-1}(b) = a \)[/tex].
4. Apply the Inverse: Using the given information [tex]\( f(18) = 2 \)[/tex]:
- We identify [tex]\( a = 18 \)[/tex] and [tex]\( b = 2 \)[/tex].
- Therefore, [tex]\( f^{-1}(2) \)[/tex] must be equal to 18 because the inverse function returns the original input for a given output.
Thus, the solution is:
[tex]\[ f^{-1}(2) = 18 \][/tex]
