График показательной функции \(f\left( x \right) = {a^x} + b\) проходит через точки \(\left( {0; — 2} \right)\) и \(\left( {4;1} \right)\). Следовательно:
\(\left\{ {\begin{array}{*{20}{c}}{ — 2 = {a^0} + b}\\{1 = {a^4} + b}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{b = — 3\,\,\,\,\,\,}\\{1 = {a^4} + b}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,{a^4} = 4\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,a = \sqrt 2 .\)
Таким образом: \(f\left( x \right) = {\sqrt 2 ^x} — 3\) и \(f\left( {10} \right) = {\sqrt 2 ^{10}} — 3 = 29.\)
Ответ: 29.