Which of these is the absolute value parent function?
A. [tex]$f(x)=x^2$[/tex]
B. [tex]$f(x)=|x|$[/tex]
C. [tex]$f(x)=2^x$[/tex]
D. [tex]$f(x)=x$[/tex]
Answer (1)
The absolute value parent function is a function that returns the non-negative value of its input.
Option A, f ( x ) = x 2 , is a quadratic function.
Option B, f ( x ) = ∣ x ∣ , is the absolute value function.
Option C, f ( x ) = 2 x , is an exponential function.
Option D, f ( x ) = x , is a linear function.
Therefore, the absolute value parent function is f ( x ) = ∣ x ∣ .
Explanation
Understanding the Problem The question asks us to identify the absolute value parent function from a list of options. Let's analyze each option to determine which one represents the absolute value parent function.
Definition of Absolute Value Function The absolute value function, denoted as f ( x ) = ∣ x ∣ , returns the non-negative value of x . In other words, if x is positive or zero, the function returns x , and if x is negative, the function returns − x . This can be defined piecewise as: f ( x ) = { x , if x ≥ 0 − x , if x < 0
Analyzing the Options Now, let's examine the given options:
A. f ( x ) = x 2 : This is a quadratic function, specifically the square function. It's not the absolute value function.
B. f ( x ) = ∣ x ∣ : This is the absolute value function, as discussed above.
C. f ( x ) = 2 x : This is an exponential function. It's not the absolute value function.
D. f ( x ) = x : This is a linear function. It returns x itself, without considering its sign. It's not the absolute value function.
Identifying the Correct Option Based on our analysis, the absolute value parent function is f ( x ) = ∣ x ∣ .
Examples
The absolute value function is used in many real-world applications, such as calculating distances or errors. For example, if you want to know the difference between a predicted value and an actual value, you can use the absolute value function to ensure that the difference is always positive, regardless of whether the prediction was higher or lower than the actual value. This is useful in fields like statistics, engineering, and finance, where it's important to quantify the magnitude of errors or deviations without considering their direction.
