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Which of the following is the parent function of all absolute value functions?

A. [tex]f(x) = 3x[/tex]

B. [tex]f(x) = |x|[/tex]

C. [tex]f(x) = 2|x|[/tex]

D. [tex]f(x) = x^2[/tex]

Sagot :

GinnyAnswer
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]
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