Final answer:
One example of a pair of functions that meet Rod's criteria are f(x) = -2 + 3x (rational) and g(x) = 3sin(x) (trigonometric). The functions have been calculated and explained step-by-step to meet each of Rod's criteria.
Step-by-step explanation:
Rod is looking for a pair of functions that meet certain criteria. Let's go through each criterion:
One function is rational and has a y-intercept at -2: One example of this could be the function f(x) = -2 + 3x. This function is rational and has a y-intercept at -2.
The other function is trigonometric and does not include the cosine function: One example of this could be the function g(x) = 3sin(x). This function is trigonometric and does not include the cosine function.
One function contains the digit '3' and the other does not: In the examples provided, f(x) contains the digit '3' and g(x) does not.
Both functions have an instantaneous rate of change of 1.23 at x = 2: To find the functions that satisfy this criterion, we need to take the derivative of each function and evaluate it at x = 2. For f(x), f'(x) = 3. The instantaneous rate of change of f(x) at x = 2 is 3. For g(x), g'(x) = 3cos(x), which is not equal to 1.23 at x = 2. Therefore, the functions f(x) = -2 + 3x and g(x) = 3sin(x) meet all of Rod's criteria.
The two functions intersect at x = 2: To verify this, we can evaluate both functions at x = 2. For f(x), f(2) = -2 + 3(2) = 4. For g(x), g(2) = 3sin(2) = 2.61 (approximate value). The two functions intersect at x = 2, as f(2) = g(2) = 4 and 2.61 respectively.