The family of functions plays a central role in the design and effectiveness of function-correcting codes. By focusing on a well-defined family of functions, function-correcting codes can be constructed with minimal length while still ensuring full error detection and correction within that family. In this work, we explore the concept of locally (λ,ρ)-functions forb-symbol read channels and investigate the optimal redundancy of the corresponding function-correctingb-symbol codes (FCBSC) by introducing the notions of locally (λ,ρ,b)-functions. First, we discuss the values ofλandρfor which a function can be considered as a locally (λ,ρ)-function inb-symbol metric. The findings improve some known results in the Hamming metric and present several new results in theb-symbol metric. Then we investigate the optimal redundancy of (f,t)-FCBSCs for locally (λ,ρ,b)-functions. We establish a recurrence relation between the optimal redundancy of (f,t)-function-correcting codes for the (b+ 1)-symbol read andb-symbol read channels. We present an upper bound on the optimal redundancy of (f,t)-function-correctingb-symbol codes for general locally (λ,ρ, b)-functions by associating it to the minimum achievable length ofb-symbol error-correcting codes and traditional Hamming-metric codes, given a fixed number of codewords and a specified minimum distance. We derive some explicit upper bounds on the redundancy of (f,t)-function-correctingb-symbol codes for locally (λ,2t,b)-functions. Moreover, for the case whereb= 1, we show that a locally (3,2t,1)-function achieves the optimal redundancy of 3t. Additionally, we explicitly investigate the locality and optimal redundancy of FCBSCs for theb-symbol weight function and weight distribution function forb≥ 1.