Final answer:
To find f(-0.5) for the exponential function where f(-3.5) = 11 and f(1.5) = 55, determine the function's base and constant by solving a system of equations, and then use the formula f(x) = ab^x.
Step-by-step explanation:
To find the value of f(-0.5), given an exponential function where f(-3.5) = 11 and f(1.5) = 55, we first need to determine the base and constant of the function. We can do this by setting up a system of equations based on the exponential function formula, f(x) = ab^x, where a is the initial value and b is the base of the exponential function.
By substituting the given points into the exponential function formula, we get two equations:
11 = ab^{-3.5}
55 = ab^{1.5}
Dividing equation (2) by equation (1) to eliminate a gives us an equation that helps us find b. Once we have both a and b, we can calculate f(-0.5) by substituting -0.5 for x in the formula f(x) = ab^x.
We can use logarithms to help solve for a and b. To simplify calculations and to avoid mistakes, it's beneficial to use the natural logarithm, as it will cancel out the exponential when you apply it on both sides of the equations.
After finding a and b, the final step is to evaluate f(-0.5) using the exponential function formula and rounding the result to the nearest hundredth.