Determine the value of the unknown, a given that the functions defined by f(x) = -5(x-3)^2 +8 and g(x) = log_3(x-1)+a intersect at x=2. Show your work for full marks.
a is the unknown
substituting ... -5(2 - 3)^2 + 8 = log_3(2 - 1) + a
-5 (1) + 8 = log_3(1) + a
3 = 0 + a
math is hard-
To find the value of the unknown, a, given that the functions f(x) and g(x) intersect at x = 2, we need to set up equations for the two functions and then solve for a.
Given functions:
f(x) = -5(x-3)^2 + 8
g(x) = log3(x-1) + a
To find the point of intersection, we set the two functions equal to each other:
-5(x-3)^2 + 8 = log3(x-1) + a
Now, we substitute x = 2 into this equation and solve for a.
-5(2-3)^2 + 8 = log3(2-1) + a
-5(-1)^2 + 8 = log3(1) + a
-5 + 8 = 0 + a
3 = a
Therefore, the value of the unknown, a, is 3.
Explanation of how to solve:
1. Start by setting the two given functions equal to each other.
2. Replace all occurrences of x with 2, as we need to find the intersection at x = 2.
3. Solve the resulting equation to find the value of a.