To determine the type of parent function for [tex]\( f(x) = x \)[/tex], let’s analyze the common types of parent functions and their general forms:
A. Reciprocal Function
- The general form of a reciprocal function is [tex]\( f(x) = \frac{1}{x} \)[/tex].
- Reciprocal functions are not linear; they have a hyperbolic curve and are undefined at [tex]\( x = 0 \)[/tex].
B. Linear Function
- The general form of a linear function is [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- A specific case of the linear function where the slope is 1 and the y-intercept is 0 is [tex]\( f(x) = x \)[/tex].
C. Quadratic Function
- The general form of a quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex].
- Quadratic functions create a parabolic curve and have a degree of 2, meaning [tex]\( x \)[/tex] is squared.
D. Absolute Value Function
- The general form of an absolute value function is [tex]\( f(x) = |x| \)[/tex].
- Absolute value functions create a V-shaped graph and are characterized by taking the absolute value of [tex]\( x \)[/tex].
Given these forms, let’s identify which category [tex]\( f(x) = x \)[/tex] belongs to:
1. [tex]\( f(x) = x \)[/tex] matches the form of a linear function [tex]\( mx + b \)[/tex] with [tex]\( m = 1 \)[/tex] and [tex]\( b = 0 \)[/tex].
2. It does not involve a reciprocal, thus it is not a reciprocal function.
3. It does not involve the square of [tex]\( x \)[/tex], so it is not a quadratic function.
4. It does not involve the absolute value, so it is not an absolute value function.
Therefore, the parent function [tex]\( f(x) = x \)[/tex] is a linear function.
Thus, the correct type is:
B. Linear
