OG详解-OG9 数学2 Q59

The correct answer is -19. It’s given that function f is defined by f(x) = a^x+b, where a and b are constants and a>0. It’s also given that the graph of y = f(x) in the xy-plane has a y-intercept at (0, -25) and passes through the point (2, 23).  Since the graph has a y-intercept at (0, -25), f(0)=-25. Substituting 0 for x in the given equation yields f(0) = a⁰ +b, or f(0) = 1 +b, and substituting -25 for f(0) in this equation yields -25 =1+b. Subtracting 1 from each side of this equation yields -26 = b. Substituting -26 for b in the equation f(x) = a^x+b yields f(x) = a^x -26. Since the graph also passes through the point (2, 23), f(2) = 23. Substituting 2 for x in the equation f(x) = a^x -26 yields f(2) = a² -26, and substituting 23 for f(2) yields 23 = a² -26. Adding 26 to each side of this equation yields 49 = a². Taking the square root of both sides of this equation yields ±7 = a. Since it’s given that a>0, the value of a is 7. It follows that the value of a+b is 7-26, or -19.