To determine the input, or independent variable in the function [tex]\( f(x) = 10 - x^3 \)[/tex], let's analyze the components of the function.
A function is a relation that assigns to each element [tex]\( x \)[/tex] in a set of inputs exactly one element [tex]\( f(x) \)[/tex] in a set of possible outputs. In the given function:
[tex]\[ f(x) = 10 - x^3 \][/tex]
1. [tex]\( x \)[/tex] is the variable that you can choose or set. It is the input to the function.
2. [tex]\( f(x) \)[/tex] represents the output of the function and depends on the value of [tex]\( x \)[/tex].
3. Sometimes, the output of a function is also denoted as [tex]\( y \)[/tex], meaning [tex]\( y = f(x) \)[/tex], but in this case, we are provided with [tex]\( f(x) \)[/tex] directly.
From the definition of the function [tex]\( f(x) \)[/tex], it is clear that the value of [tex]\( f(x) \)[/tex] (the output) is determined by the value of [tex]\( x \)[/tex]. Therefore, the independent variable, or the input variable, is [tex]\( x \)[/tex].
Thus, the answer is:
[tex]\( x \)[/tex]
