Determining the Range of the Function f(x) |x-1| - |x-5|

In the realm of mathematical analysis, a fundamental concept is the ldquo;rangerdquo; of a function, which refers to the set of all possible output values for a given set of input values. In this article, we aim to explore the range of the function ( f(x) |x-1| - |x-5| ). By understanding the behavior of this function across various intervals of ( x ), we can determine its complete range.

Step 1: Identifying Critical Points

The key to analyzing the function ( f(x) |x-1| - |x-5| ) lies in identifying the critical points at which the expressions inside the absolute values change. These critical points are ( x 1 ) and ( x 5 ). Based on these points, we can divide the real number line into three intervals and analyze the function separately in each interval.

Interval 1: ( x

For ( x

|x-1|  1-x,  |x-5|  5-x

Substituting these into the function, we get:

f(x)  (1-x) - (5-x)  1-x-5 x  -4

Therefore, for ( x f(x) -4.

Interval 2: ( 1 leq x

For ( 1 leq x

|x-1|  x-1,  |x-5|  5-x

Substituting these into the function, we get:

f(x)  (x-1) - (5-x)  x-1-5 x  2x - 6

This is a linear function that can be evaluated at the endpoints:

At ( x 1 ): ( f(1) 2*1 - 6 -4 )

At ( x 5 ): ( f(5) 2*5 - 6 4 )

Therefore, in the interval ( 1 leq x

Interval 3: ( x geq 5 )

For ( x geq 5 ), both ( x-1 ) and ( x-5 ) are non-negative, so:

|x-1|  x-1,  |x-5|  x-5

Substituting these into the function, we get:

f(x)  (x-1) - (x-5)  x-1-x 5  4

Therefore, for ( x geq 5 ), f(x) 4.

Step 3: Combining Results

By combining the results from the three intervals, we can summarize the function's behavior as follows:

For ( x

For ( 1 leq x

For ( x geq 5 ), ( f(x) 4 ).

Therefore, the complete range of the function ( f(x) |x-1| - |x-5| ) is:

Range: ([-4, 4])

Conclusion

To conclude, the function ( f(x) |x-1| - |x-5| ) has a range that spans from (-4) to (4). This result can be verified through the analysis of the function in three distinct intervals, each representing a different piecewise behavior of the function.

For a more detailed interactive exploration, you can experiment with different values of ( x ) using graphing software or online tools to confirm the insight provided by the piecewise analysis.