To determine the conditions under which the function [tex]\( f(x) = mx + b \)[/tex] has an inverse function, we need to consider the properties necessary for a function to have an inverse.
1. Bijectivity: For a function to have an inverse, it must be bijective. Being bijective means the function must be both injective (one-to-one) and surjective (onto).
2. Injectivity: A function is injective if every element of the function's codomain is mapped by at most one element of its domain. For a linear function [tex]\( f(x) = mx + b \)[/tex], injectivity is achieved if and only if the slope [tex]\( m \)[/tex] is not zero. If [tex]\( m \)[/tex] were zero, the function would be a horizontal line, [tex]\( f(x) = b \)[/tex], which maps every [tex]\( x \)[/tex] to the same value [tex]\( b \)[/tex]. Consequently, it would not be injective because different values of [tex]\( x \)[/tex] would map to the same [tex]\( f(x) \)[/tex].
3. Surjectivity: A linear function [tex]\( f(x) = mx + b \)[/tex] is inherently surjective for all real numbers because it can produce any real number output by appropriately choosing [tex]\( x \)[/tex], provided [tex]\( m \neq 0 \)[/tex].
As we've established, the critical factor in determining whether [tex]\( f(x) = mx + b \)[/tex] has an inverse is whether it is injective. This is determined by the slope [tex]\( m \)[/tex].
Therefore, for the function [tex]\( f(x) = mx + b \)[/tex] to have an inverse, it must be true that:
[tex]\[ m \neq 0 \][/tex]
So, the correct statement is:
[tex]\[ m \neq 0 \][/tex]
