IF
f(x) = {(3,5), (2,4), (1,7)}
h(x)= {(3,2), (4,3), (1,6)}
determine each of the following:
(f+h)(1)=
how do I add f and h?
(f o h)(3)=
how would I multiply f and h?
f^-1(x)=
how would I find the inverse of f?
1/f(x)=
how would I do that?
f(1) = 7 and h(1) = 6
(f + h)(x) is by definition f(x) + h(x)
(f o h)(x) is by definition f(h(x))
h(3) = 2
and f(h(3)) = f(2) = 4
So (foh)(3) = 4
f^(-1)(f(x)) = x by definition.
f(x) = {(3,5), (2,4), (1,7)}
If you take, e.g., x = 3, then you see that
f^(-1)(5) = 3. So, you see that you just need to interchanbge the numbers:
f^(-1)(x) = {(5,3), (4,2), (7,1)}
To find 1/f(x), you would take the reciprocal of each y-value in f(x). Here's the process:
f(x) = {(3,5), (2,4), (1,7)}
1/f(x) = {(3,1/5), (2,1/4), (1,1/7)}
So, 1/f(x) would be {(3,1/5), (2,1/4), (1,1/7)}.
Remember, humor is my specialty, not math!
To add f and h, you simply add the corresponding y-values for each x-value. In this case:
(f + h)(1) = f(1) + h(1) = 7 + 6 = 13
To multiply f and h, you need to find the composition of the two functions. This means you apply the function h first, and then apply f to the result. In this case, you have:
(f o h)(3) = f(h(3)) = f(2) = 4
To find the inverse of f, you need to interchange the x and y values in the given set of points. So the inverse of f is:
f^(-1)(x) = {(5,3), (4,2), (7,1)}
To find 1/f(x), you need to take the reciprocal of each y-value in the given set of points. So:
1/f(x) = {(3,1/5), (4,1/4), (7,1/7)}
To add f and h, you need to add the corresponding y-values for each given x-value. In this case, (f+h)(1) would be the sum of f(1) and h(1). Since f(1) is 7 and h(1) is 6, you add them together: 7 + 6 = 13. Therefore, (f+h)(1) = 13.
To multiply f and h, you need to compose the two functions. In other words, evaluate f at the values of h. For (f o h)(3), you need to find h(3) first, which is 2. Then, you evaluate f at 2: f(2) = 4. Therefore, (f o h)(3) = 4.
To find the inverse of f, you need to interchange the x and y values in the original function. In this case, f(x) = {(3,5), (2,4), (1,7)}. So, the inverse function f^(-1)(x) will have the x-values as its y-values and vice versa. Therefore, f^(-1)(x) = {(5,3), (4,2), (7,1)}.
To compute 1/f(x), you need to find the reciprocal of each y-value in the function f(x). In this case, f(x) = {(3,5), (2,4), (1,7)}. So, the reciprocal of 5 is 1/5, the reciprocal of 4 is 1/4, and the reciprocal of 7 is 1/7. Therefore, 1/f(x) = {(3,1/5), (2,1/4), (1,1/7)}.