Understanding the Range of the Function ( sqrt{3-x} )

To determine the range of the function ( f(x) sqrt{3-x} ), we need to explore the behavior of the function based on its domain and how it behaves within that domain.

Domain and Behavior of the Function

The expression inside the square root, ( 3-x ), must be non-negative for the function to have real values. Thus, the domain of ( f(x) ) is all ( x ) such that:

[ 3-x geq 0 ]

Solving for ( x ), we get:

[ x leq 3 ]

Therefore, the domain of ( f(x) ) is ( (-infty, 3] ).

Range of the Function

The range of the function represents the set of all possible output values (y-values) that ( f(x) ) can take. Let's analyze how the function behaves within the domain.

Behavior at Specific Points

At ( x 3 ), we have:

[ f(3) sqrt{3-3} sqrt{0} 0 ]

This tells us that ( y 0 ) is in the range of the function.

For ( x > 3 ), the expression inside the square root becomes negative, and the square root of a negative number is undefined in the set of real numbers. Therefore, for ( x > 3 ), ( f(x) ) is undefined in the reals.

Other Key Points

At ( x -1 ), we have:

[ f(-1) sqrt{3-(-1)} sqrt{4} 2 ]

This tells us that ( y 2 ) is a possible value in the range of the function.

For ( x

The minimum value of ( f(x) ) is 0, which occurs at the boundary of the domain, ( x 3 ).

Graphical Analysis

To visualize the function and confirm the range, we can graph it using Desmos:

From the graph, we can see that the function starts at ( y 0 ) when ( x 3 ) and increases to a maximum value of 2 when ( x 0 ). For values of ( x ) less than 0, the graph continues to increase without bound.

Conclusion

The range of the function ( f(x) sqrt{3-x} ) is the interval [0, 2]. This means the function can take any value from 0 to 2, inclusive.

This analysis shows the importance of understanding the domain and how the function behaves within it. By examining key points and graphing the function, we can determine the complete range of the function.