Question

Rod says that he is thinking of a pair of functions that have all of the following characteristics: a. One of the two functions is rational, and has a y-intercept at -2 b. The other of the two functions is trigonometric, and does not include the cosine function c. One of the two functions contains the digit "3" and the other does not. d. Both have an instantaneous rate of change of 1.23 (rounded to two decimal places) at x = 2 e. The two functions intersect at x = 2 Provide one example of a pair of functions that meet all of Rod’s criteria. Explain your thought process in making the functions, a screenshot, and calculations to verify each criterion.

Answer 1

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.

Answer 2

A pair of functions that satisfy Rod's criteria are the rational function f(x) = 1.23x - 2 and the trigonometric function g(x) = 1.23x - 1.77 (a sine function not included). Both functions have a y-intercept, contain/not contain the digit '3', an instantaneous rate of change of 1.23 at x = 2, and intersect at x = 2.

Step-by-step explanation:

According to the criteria provided by Rod, we need to find a pair of functions that satisfy several conditions. Let's consider each condition:

• The rational function with a y-intercept at -2 can be f(x) = 1.23x - 2. This satisfies the y-intercept requirement and the rate of change at x = 2.
• The trigonometric function that is not cosine might be the sine function; so, to have the digit '3' and an instantaneous rate of change of 1.23 at x = 2, we can use g(x) = 1.23x - 1.77 (assuming the '3' must be in the slope and not in the intercept).
• We confirm that g(x) does not include the cosine function and the rational function f(x) does not contain the digit '3'.
• Both functions have an instantaneous rate of change of 1.23 at x = 2, since both slopes are 1.23.
• Both functions intersect at x = 2 because f(2) = g(2) = 1.23(2) - 2 = 1.23(2) - 1.77.

To verify each criterion, one can calculate the value of the functions at x = 2 and check the slopes at this point by differentiation. For this example, differentiation is not necessary since the slopes are given by the coefficients of x (1.23).