(with different domain) is a different function, even though the formulas defining * f* and * g* agree. Similarly, a function with a different codomain is also a different function. Nevertheless, many authors do not specify the domain and codomain, especially if these are clear from context. So in this example many just write * f* (* x* ) = 4 − * x* . Sometimes, the maximal possible domain within a larger set implied by context is implicitly understood: a formula such as
f
(
x
)
=
x
2
−
5
x
+
6
{\displaystyle \textstyle f(x)={\sqrt {x^{2}-5x+6}}}
may mean that the domain of * f* is the set of real numbers * x* where the square root is defined (in this case * x* ≤ 2 or * x* ≥ 3 ). However, in a different context, this expression might refer to a complex-valued function
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
.

The prevalence of dementia is increasing with expansion of the older adult population. In the absence of effective therapy, preventive approaches are essential to address this public health problem. Blueberries contain polyphenolic compounds, most prominently anthocyanins, which have antioxidant and anti-inflammatory effects. In addition, anthocyanins have been associated with increased neuronal signaling in brain centers, mediating memory function as well as improved glucose disposal, benefits that would be expected to mitigate neurodegeneration. This study investigated the effects of daily consumption of wild blueberry juice in a sample of nine older adults with early memory changes. At 12 weeks, improved paired associate learning (p = ) and word list recall (p = ) were observed. In addition, there were trends suggesting reduced depressive symptoms (p = ) and lower glucose levels (p = ). We also compared the memory performances of the blueberry subjects with a demographically matched sample who consumed a berry placebo beverage in a companion trial of identical design and observed comparable results for paired associate learning. The findings of this preliminary study suggest that moderate-term blueberry supplementation can confer neurocognitive benefit and establish a basis for more comprehensive human trials to study preventive potential and neuronal mechanisms.

A common solution is to compute a fixed hash function with a very large range (say, 0 to 2 32 − 1), divide the result by * n* , and use the division's remainder . If * n* is itself a power of 2, this can be done by bit masking and bit shifting . When this approach is used, the hash function must be chosen so that the result has fairly uniform distribution between 0 and * n* − 1, for any value of * n* that may occur in the application. Depending on the function, the remainder may be uniform only for certain values of * n* , . odd or prime numbers .