To solve this problem of matching the given function [tex]\( f(x) = x^2 \)[/tex] with its name, let's analyze the given options:
1. Greatest integer function: This function, often denoted as [tex]\( \lfloor x \rfloor \)[/tex], rounds down a real number to the nearest integer. It is not [tex]\( x^2 \)[/tex].
2. Absolute value function: This function is denoted as [tex]\( |x| \)[/tex] and returns the non-negative value of [tex]\( x \)[/tex]. It is a different function from [tex]\( x^2 \)[/tex].
3. Squaring function: This option is directly related to the function [tex]\( f(x) = x^2 \)[/tex]. As the name suggests, squaring a number means multiplying the number by itself, which is exactly what [tex]\( f(x) = x^2 \)[/tex] does.
4. Cubic function: This function is of the form [tex]\( f(x) = x^3 \)[/tex]. It involves raising [tex]\( x \)[/tex] to the third power, not the second power.
5. Linear function: This function is of the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. It does not involve squaring [tex]\( x \)[/tex].
Thus, the correct match for the function [tex]\( f(x) = x^2 \)[/tex] is squaring function.
