Consider the function f ( x )= a ^ [ x ]+ x -1 [ x ]+ x where [ ] denotes the greatest integer function. What is _ x 0⁻ f(x) equal to?

Step-by-step solution

aligned _ x 0⁻ f(x) & = _ h 0 [ a^ [0-h]+(0-h) -1 [0-h]+(0-h) ] & = _ h 0 a^ [-h]-h -1 [-h]+(-h) & = _ h 0 a^ -1-h -1 -1-h & = a⁻¹⁻⁰-1 -1-0 = a⁻¹-1 -1 & = _ h 0⁻ f(x)= (1-a⁻¹ ) aligned