Which of the following are inverse functions?
1. Arcsin x and sin x
2. cos^-1 x and cos x
3. csc x and sin x
4. e^x and ln x
5. x^2 and +/- sqrt x
6. x^3 and cubic root of x
7. cot x and tan x
8. sin x and cos x
9. log x/3 and 3^x
I believe the answers are 2, 4, 6, and 9, but how come not the rest, especially csc/sin and cot/tan?
Thank you very much!
pick any x
perform the first operation
perform the second operation on that answer.
did I get back the original x ?
cscx and sinx, cotx and tanx are reciprocal functions, not inverses
my choices of inverse are:
1,2,4,6
(how did you figure #9 represents an inverse ???)
To determine whether two functions are inverse functions, we need to check if they "undo" each other. In other words, if applying one function and then the other function gives us back the original value. Let's go through each pair of functions to see if they are inverses:
1. Arcsin x and sin x:
To check if these are inverses, we need to apply sin x to the result of Arcsin x and see if we get back the original value. However, sin x is not an invertible function because it does not satisfy the horizontal line test. Therefore, Arcsin x and sin x are not inverse functions.
2. cos^-1 x and cos x:
To check if these are inverses, we apply cos x to the result of cos^-1 x. Using the identity cos(cos^-1(x)) = x, we can see that they "undo" each other, resulting in x. Therefore, cos^-1 x and cos x are inverse functions.
3. csc x and sin x:
To determine if these are inverses, we apply sin x to the result of csc x. However, csc x is not an "undo" function for sin x because csc x = 1/sin x does not give us back the original value. Therefore, csc x and sin x are not inverse functions.
4. e^x and ln x:
To check if these are inverses, we apply ln x to the result of e^x. By applying the natural logarithm to e^x, we obtain ln(e^x) = x. Therefore, e^x and ln x are inverse functions.
5. x^2 and +/- sqrt x:
To determine if these are inverses, we apply sqrt x to the result of x^2. However, sqrt x is not the inverse of x^2 because it only gives us the positive square root, not both the positive and negative roots. Therefore, x^2 and +/- sqrt x are not inverse functions.
6. x^3 and cubic root of x:
To check if these are inverses, we apply the cubic root function to the result of x^3. Mathematically, (cubic root of x^3) = x. Therefore, x^3 and the cubic root of x are inverse functions.
7. cot x and tan x:
To determine if these are inverses, we apply tan x to the result of cot x. However, cot x is not an invertible function because it does not satisfy the horizontal line test. Therefore, cot x and tan x are not inverse functions.
8. sin x and cos x:
To check if these are inverses, we apply cos x to the result of sin x. However, sin x is not the inverse of cos x because it only gives us the value of the sine function, not the input itself. Therefore, sin x and cos x are not inverse functions.
9. log x/3 and 3^x:
To determine if these are inverses, we apply 3^x to the result of log x/3. Using the logarithmic property log(base 3)(3^x) = x, we can see that they "undo" each other, resulting in x. Therefore, log x/3 and 3^x are inverse functions.
In summary, the inverse functions from the given list are: cos^-1 x and cos x, e^x and ln x, x^3 and cubic root of x, and log x/3 and 3^x. The rest of the functions are not inverses of each other.