To determine which of the given functions has values of its range decrease as the values in its domain increase, we need to look at their behavior and properties. Specifically, we are interested in the slope of each function. If the function is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope:
1. A positive slope ([tex]\( m > 0 \)[/tex]) indicates that the function's value increases as [tex]\( x \)[/tex] (the domain) increases.
2. A negative slope ([tex]\( m < 0 \)[/tex]) indicates that the function's value decreases as [tex]\( x \)[/tex] (the domain) increases.
3. A zero slope ([tex]\( m = 0 \)[/tex]) indicates that the function's value remains constant, regardless of [tex]\( x \)[/tex].
Let's examine each option:
### Option A: [tex]\( y = \frac{1}{3}x \)[/tex]
- The slope ([tex]\( m \)[/tex]) is [tex]\( \frac{1}{3} \)[/tex], which is positive.
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will increase.
- This does not match our requirement.
### Option B: [tex]\( y = -2x + 4 \)[/tex]
- The slope ([tex]\( m \)[/tex]) is [tex]\(-2\)[/tex], which is negative.
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will decrease.
- This matches our requirement.
### Option C: [tex]\( y = -\frac{1}{3} \)[/tex]
- This is a constant function where [tex]\( y \)[/tex] is always [tex]\(-\frac{1}{3} \)[/tex], regardless of [tex]\( x \)[/tex].
- There's no increase or decrease in [tex]\( y \)[/tex] as [tex]\( x \)[/tex] changes.
- This does not match our requirement.
### Option D: [tex]\( y = 2x + 5 \)[/tex]
- The slope ([tex]\( m \)[/tex]) is [tex]\( 2 \)[/tex], which is positive.
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will increase.
- This does not match our requirement.
After examining all the options, we can conclude that the function that has the values of its range decrease as the values in its domain increase is:
Option B: [tex]\( y = -2x + 4 \)[/tex].
