Helppp........................

Question

Using interval notation, determine the largest domain over which the given function is one‑to‑one. Then, provide the equation for the inverse of the function that is restricted to that domain.

If two equally large domains exist over which the given function is one‑to‑one, you may use either domain. However, be certain that the equation for the inverse function you submit is appropriate for the particular domain you choose.

𝑓(𝑥)=1/√x^2(+5)

domain:

𝑓 ^−1(𝑥)=

Answer 1

Although f(x) = 1;//sqrt(x²+5) is a continuous function, it is not one-to-one in an unlimited domain because it is symmetric about the y-axis. Therefore, the domain must be limited to x in [0,∞)

The technique for finding the inverse function is to switch x and y in the function and solve for y:

1/(sqrt(y²+5) = x

sqrt(y²+5) = 1/x

y²+5 = 1/x²

y = sqrt(1/x² - 5) requires, x = 0 and 1/x² - 5 > 0 or -1/sqrt(5)<x<1/sqrt(5)

The original function did not have negative values, so we also have to stipulate that x is greater or equal to 0 also in order to maintain the one-to-one correspondence for the inverse as well: x in (0, 1/sqrt(5)]