To determine the parent function of all absolute value functions, let's review the options provided:
1. [tex]\(f(x) = 3x\)[/tex]
2. [tex]\(f(x) = |x|\)[/tex]
3. [tex]\(f(x) = 2|x|\)[/tex]
4. [tex]\(f(x) = x^2\)[/tex]
### Step-by-Step Analysis:
1. Option [tex]\(f(x) = 3x\)[/tex]:
- This function represents a straight line with a slope of 3. It is a linear function but not an absolute value function.
2. Option [tex]\(f(x) = |x|\)[/tex]:
- This is the standard form of an absolute value function. Absolute value functions have a characteristic 'V' shape and can be written in the form [tex]\(f(x) = a|x - h| + k\)[/tex], where [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] are constants. The simplest or "parent" form of this function is [tex]\(f(x) = |x|\)[/tex].
3. Option [tex]\(f(x) = 2|x|\)[/tex]:
- This function is still an absolute value function but is scaled vertically by a factor of 2. Hence, it is a transformation of the parent function [tex]\(f(x) = |x|\)[/tex], not the parent function itself.
4. Option [tex]\(f(x) = x^2\)[/tex]:
- This function represents a parabola, which is a quadratic function. It is not an absolute value function.
Given these analyses, the parent function of all absolute value functions is:
[tex]\[f(x) = |x|\][/tex]
So, the correct answer is:
[tex]\[f(x) = |x|\][/tex]
