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!
To determine which of the given functions are inverse functions, we need to understand what inverse functions are. Inverse functions are pairs of functions that "undo" each other. When composed together, they result in the identity function. In other words, given a function f(x) and its inverse function g(x), if we evaluate f(g(x)) or g(f(x)), it will always yield x.
Now, let's analyze each pair of functions:
1. Arcsin x and sin x: These functions are not inverses of each other because arcsin (sin x) is only defined for values of x in the range of [-π/2, π/2]. Thus, applying arcsin to the result of sin x will not always yield x.
2. cos^-1 x and cos x: These functions are indeed inverse functions. Applying cos^-1(cos x) or cos(cos^-1 x) will always give us x. The inverse cos^-1(x) is commonly denoted as acos(x).
3. csc x and sin x: These functions are not inverses of each other. The inverse of sin x is arcsin x, not csc x. Cosecant (csc x) is the reciprocal of sin x, so applying csc(sin x) or sin(csc x) will not always yield x.
4. e^x and ln x: These are indeed inverse functions. Applying e^(ln x) or ln(e^x) will always result in x.
5. x^2 and +/- sqrt(x): These functions are not inverses of each other. The inverse of x^2 is the principal square root of x, denoted as sqrt(x), not +/- sqrt x. Applying sqrt(x^2) or (x^2)^(1/2) will only yield the positive square root of x, not the negative one.
6. x^3 and cubic root of x: These functions are inverses of each other. Applying (cubic root of x)^3 or (x^3)^(1/3) will always give us x.
7. cot x and tan x: These functions are not inverses of each other. The inverse of tan x is arctan x, not cot x. Cotangent (cot x) is the reciprocal of tan x, so applying cot(tan x) or tan(cot x) will not always yield x.
8. sin x and cos x: These functions are not inverses of each other. However, they are closely related through trigonometric identities, such as sin^2x + cos^2x = 1.
9. log x/3 and 3^x: These functions are inverses of each other. Applying log(3^x)/3 or 3^(log x/3) will result in x.
Therefore, the correct choices for inverse functions are 2 (cos and cos^-1), 4 (e^x and ln x), 6 (x^3 and cubic root of x), and 9 (log x/3 and 3^x). The rest of the pairs are not inverses of each other due to different domains, ranges, or reciprocal relationships.