Given the following values for f(t):
t : f(t)
3 : -7
6 : 6
9 : 19
12 : 6
15 : -7
18 : 6
a) What is the period, frequency, midline, amplitude, maximum, and minimum?
b)Find a possible formula for f(t)
c)Find the domain and range of f(t)
d)Find a formula for an inverse function of f(t). (Hint: you will need to restrict the domain)What is the domain and range of the inverse function?
sketch it
It goes from -7 to -7, one period, in (15-3) = 12 second period
frequency = 1/period = 1/12
midline = (19-7)/2 = 6
single amplitude = 19-6 = 13 so double amplitude = 26
max = 19. min = -7
domain from 3 to 18 unless you let it go on the same shape from -oo to +oo
range from -7 to + 19
There is no inverse function for the whole thing because there are multiple values of x for the same y in the original, therefore multiple values of y for the same x in the invers
Therefore you will have to split it up in the original to the section from x = 3 o 9, then from 9 to 15 , then from 15 to 18 and it is a nice straight line in each.
a) To find the period, frequency, midline, amplitude, maximum, and minimum of the given function, we can observe the pattern in the given values for f(t):
Period: The period is the horizontal distance between two consecutive cycles or repetitions of the function. Looking at the values of t, we can see that the pattern repeats every 6 units. So the period of f(t) is 6.
Frequency: The frequency is the reciprocal of the period, which represents how many cycles occur per unit. In this case, since the period is 6, the frequency is 1/6.
Midline: The midline is the horizontal line that the function oscillates around. To find the midline, we can take the average of the maximum and minimum values of f(t). In this case, the maximum is 19 and the minimum is -7. The average of these is (19 + (-7))/2 = 6. So the midline is y = 6.
Amplitude: The amplitude is the distance from the midline to the maximum or minimum. In this case, since the midline is 6, the amplitude is |6 - (-7)| = 13.
Maximum: The maximum value of f(t) is 19.
Minimum: The minimum value of f(t) is -7.
b) To find a possible formula for f(t), we can examine the pattern in the given values. Looking at the values, we can see that f(t) alternates between two values, 6 and -7, every 6 units. We can express this pattern mathematically using a piecewise function:
f(t) = {
6, if t mod 6 < 3,
-7, if t mod 6 >= 3,
}
Here, "t mod 6" represents the remainder when t is divided by 6. For example, if t = 15, then t mod 6 = 15 mod 6 = 3.
c) The domain of f(t) is the set of all possible input values t. In this case, the given values for t are 3, 6, 9, 12, 15, and 18. Therefore, the domain of f(t) is {3, 6, 9, 12, 15, 18}.
The range of f(t) is the set of all possible output values. Looking at the given values for f(t), we can see that the range is {-7, 6, 19}.
d) To find the inverse function of f(t), we need to switch the roles of t and f(t).
Let's call the inverse function g(t). We can express g(t) using a piecewise function based on the pattern we observed in f(t):
g(t) = {
3, if f(t) = 6,
6, if f(t) = -7,
9, if f(t) = 19,
}
Here, g(t) maps the values of f(t) back to the corresponding values of t.
The domain of g(t) will be the range of f(t), which is {-7, 6, 19}. The range of g(t) will be the domain of f(t), which is {3, 6, 9, 12, 15, 18}.