Which parent function is represented by the table?
| x | y |
| --- | --- |
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
A. [tex]f(x)=x^2[/tex]
B. [tex]f(x)=2^x[/tex]
C. [tex]f(x)=|x|[/tex]
D. [tex]f(x)=x[/tex]
Answer (1)
Test each function with the given x-values.
f ( x ) = x 2 does not match the table.
f ( x ) = 2 x does not match the table.
f ( x ) = ∣ x ∣ matches the table.
f ( x ) = x does not match the table.
The parent function represented by the table is f ( x ) = ∣ x ∣ .
Explanation
Analyzing the Problem We are given a table of x and y values and asked to determine which of the given parent functions matches the table. The table contains the following points: (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2). The possible parent functions are: A. f ( x ) = x 2 B. f ( x ) = 2 x C. f ( x ) = ∣ x ∣ D. f ( x ) = x
Testing Each Function Let's test each function with the given x-values to see which one produces the corresponding y-values.
For A, f ( x ) = x 2 :
f ( − 2 ) = ( − 2 ) 2 = 4 f ( − 1 ) = ( − 1 ) 2 = 1 f ( 0 ) = ( 0 ) 2 = 0 f ( 1 ) = ( 1 ) 2 = 1 f ( 2 ) = ( 2 ) 2 = 4 This does not match the table since f ( − 2 ) should be 2, not 4, and f ( 2 ) should be 2, not 4.
For B, f ( x ) = 2 x :
f ( − 2 ) = 2 − 2 = 4 1 f ( − 1 ) = 2 − 1 = 2 1 f ( 0 ) = 2 0 = 1 f ( 1 ) = 2 1 = 2 f ( 2 ) = 2 2 = 4 This does not match the table.
For C, f ( x ) = ∣ x ∣ :
f ( − 2 ) = ∣ − 2∣ = 2 f ( − 1 ) = ∣ − 1∣ = 1 f ( 0 ) = ∣0∣ = 0 f ( 1 ) = ∣1∣ = 1 f ( 2 ) = ∣2∣ = 2 This matches the table exactly!
For D, f ( x ) = x :
f ( − 2 ) = − 2 f ( − 1 ) = − 1 f ( 0 ) = 0 f ( 1 ) = 1 f ( 2 ) = 2 This does not match the table since f ( − 2 ) should be 2, not -2, and f ( − 1 ) should be 1, not -1.
Determining the Correct Function Only the function f ( x ) = ∣ x ∣ matches the table of values. Therefore, the parent function represented by the table is f ( x ) = ∣ x ∣ .
Examples
Understanding parent functions like f ( x ) = ∣ x ∣ is crucial in various real-world applications. For instance, consider the scenario of measuring the distance of a point from a reference point, irrespective of direction. The absolute value function helps represent this distance, as it always returns a non-negative value. This concept is used in physics to calculate displacement, in engineering to determine tolerances, and even in economics to model deviations from a target value. By recognizing and applying parent functions, we can model and analyze a wide range of phenomena in a simplified and effective manner.
