The graph A(x) has points (-3,0), (0,-2), (1,0) and (2,3).
If C(x) is defined by C(x)=A(2x+3), then what is the domain and range of C and how would the graph of C(x) look like?
Also, is there any way to find what the actual equation of A(x) is with the points listed above? Thank you for the help!
To find the domain and range of C(x), we need to consider the values of x that are valid inputs for C(x) and the corresponding outputs of C(x).
For the domain of C(x), we need to find the values of x that make the expression 2x + 3 valid. Since A(x) is given for the points (-3,0), (0,-2), (1,0), and (2,3), we can see that A(x) is defined for x = -3, 0, 1, and 2.
Now, for C(x) = A(2x + 3), we need to determine the values of 2x + 3 that correspond to the valid inputs of A(x). From earlier, we know that x = -3, 0, 1, and 2 are valid inputs for A(x).
To find the corresponding values of 2x + 3, we substitute these values into the expression:
For x = -3: 2x + 3 = 2(-3) + 3 = -6 + 3 = -3
For x = 0: 2x + 3 = 2(0) + 3 = 0 + 3 = 3
For x = 1: 2x + 3 = 2(1) + 3 = 2 + 3 = 5
For x = 2: 2x + 3 = 2(2) + 3 = 4 + 3 = 7
Therefore, the domain of C(x) is {-3, 0, 1, 2}, since these are the values of x that produce valid inputs for A(x).
To determine the range of C(x), we need to find the corresponding outputs of A(x) for the domain values we found earlier. From the points given for A(x), we can see that the corresponding outputs for x = -3, 0, 1, and 2 are 0, -2, 0, and 3, respectively.
Hence, the range of C(x) is {0, -2, 3}.
Now, let's discuss how the graph of C(x) would look like. The function C(x) = A(2x + 3) is a transformation of the graph of A(x) due to the substitution of 2x + 3 for x. This transformation involves a horizontal compression (by a factor of 1/2) and a horizontal translation to the left by 3 units.
To graph C(x), you can start by graphing A(x) using the given points (-3,0), (0,-2), (1,0), and (2,3). Once you have graphed A(x), apply the horizontal compression and the horizontal translation to obtain the graph of C(x).
Regarding finding the actual equation of A(x) using the given points, we can use the method of finding the equation of a line. Since A(x) is not explicitly described, assuming it is a linear function, we can calculate the slope using the formula:
slope (m) = (change in y) / (change in x)
Taking any two given points, let's say (1,0) and (0,-2), we have:
m = (0 - (-2)) / (1 - 0) = 2/1 = 2
So, the slope of A(x) is 2.
Using the point-slope form of the equation of a line, we can write:
y - y1 = m(x - x1)
Using the point (1,0) and the slope m = 2, we can substitute these values to find the equation of A(x):
y - 0 = 2(x - 1)
Simplifying,
y = 2(x - 1)
Therefore, the equation of A(x) is y = 2x - 2.
Hope this helps! Let me know if you have any further questions.