Decision problems vs functions
Another reason is that this is often without loss of generality, since frequently (though not always - see below) complexity of functions and decision problems are equivalent. Every decision problem can be viewed as a function whose only values are 0 and 1. Conversely, given a function $f$, there are several associated decision problems which usually have the same complexity as $f$, for example:


$\{(x,i) : $ the $i$-th bit of $f(x)$ is 1$\}$.
$\{(x,k) : f(x) \leq k\}$ (or $\geq$)


Here is an example where the complexity of function classes and their associated language classes seem to differ: $\mathsf{P}^{\mathsf{NP}[\log ]} = \mathsf{P}^{\mathsf{NP}}_{tt}$ [Wagner 1987 "Log Query Classes", Hemaspaandra's 1987 thesis, Buss &amp; Hay 1991],  but if $\mathsf{FP}^{\mathsf{NP}[\log ]} = \mathsf{FP}^{\mathsf{NP}}_{tt}$ then $\mathsf{NP} = \mathsf{RP}$ and $\mathsf{P} = \mathsf{UP}$ [Selman 1994]. 

(Here the oracle $^{\mathsf{NP}[\log]}$ means the machine gets to make $O(\log n)$ queries to any problem in $\mathsf{NP}$ (say, SAT). The notation $\mathsf{P}^{\mathsf{NP}}_{tt}$ means $\mathsf{P}$ with an $\mathsf{NP}$ oracle, but in which the oracle queries are non-adaptive: on input $x$, the if $y_i$ is the $i$-th string queried to the oracle, then $y_i$ does not depend on the answers to any previous oracle calls. Equivalently, on input $x$ the machine builds a list $y_1, \dotsc, y_m$ without querying the oracle, queries the oracle about all of the $y_i$ and gets the answers, then proceeds to compute without querying the oracle again.)