To determine which function has an inverse that is also a function, we need to analyze each given function to see if it meets the criteria for having an inverse. A function has an inverse if and only if it is bijective, which means it must be both one-to-one (injective) and onto (surjective).
Let’s consider each function one by one:
1. [tex]\( d(x) = -9 \)[/tex]:
- This is a constant function. No matter what value [tex]\( x \)[/tex] takes, [tex]\( d(x) \)[/tex] will always equal [tex]\(-9\)[/tex].
- Since all values of [tex]\( x \)[/tex] map to the same output, [tex]\( d(x) \)[/tex] is not one-to-one. Different inputs can result in the same output.
- Therefore, [tex]\( d(x) \)[/tex] does not have an inverse that is a function because it fails the one-to-one criteria.
2. [tex]\( m(x) = -7x \)[/tex]:
- This is a linear function with a slope of [tex]\(-7\)[/tex] and it is of the form [tex]\( y = ax \)[/tex] where [tex]\( a \neq 0 \)[/tex].
- Linear functions with non-zero slopes are one-to-one because every unique input [tex]\( x \)[/tex] will give a unique output [tex]\( -7x \)[/tex].
- Moreover, [tex]\( m(x) \)[/tex] covers all real numbers as [tex]\( x \)[/tex] ranges over all real numbers. So it is onto.
- Hence, [tex]\( m(x) \)[/tex] is both one-to-one and onto, meaning it is bijective and does have an inverse that is a function.
3. [tex]\( p(x) = |x| \)[/tex] (absolute value function):
- The absolute value function is not one-to-one because both a positive [tex]\( x \)[/tex] and its negative counterpart [tex]\(-x \)[/tex] yield the same output [tex]\( |x| \)[/tex].
- For example, [tex]\( p(3) = 3 \)[/tex] and [tex]\( p(-3) = 3 \)[/tex]. Different inputs producing the same output violates the one-to-one condition.
- Therefore, [tex]\( p(x) \)[/tex] does not have an inverse that is a function.
Based on this analysis, the function that has an inverse which is also a function is:
[tex]\( m(x) = -7x \)[/tex].
So, the correct answer is:
[tex]\( \boxed{2} \)[/tex]
