Answer:
5)
The inverse of the function f(x) = x^7 can be found by following these steps:
Step 1: Replace f(x) with y. The equation becomes y = x^7.
Step 2: Interchange x and y in the equation, so it becomes x = y^7.
Step 3: Solve the equation for y by taking the seventh root of both sides. This yields y = x^(1/7).
Therefore, the inverse function of f(x) = x^7 is g(x) = x^(1/7), which maps any given value of x to its seventh root.
It's important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is only positive real numbers. The domain of the original function is all real numbers, while the range is also all real numbers.
6)
To find the inverse of the function f(x) = (-2/5)x^3, we can follow these steps:
Step 1: Replace f(x) with y. The equation becomes y = (-2/5)x^3.
Step 2: Solve the equation for x in terms of y.
Multiply both sides by -5/2:
(-5/2) y = x^3
Take the cube root of both sides:
x = [(-5/2) y]^(1/3)
Step 3: Replace x with f^-1(y) to obtain the inverse function.
f^-1(y) = [(-5/2) y]^(1/3)
Therefore, the inverse function of f(x) = (-2/5)x^3 is f^-1(y) = [(-5/2) y]^(1/3).
It is important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is also all real numbers. The domain of the original function is all real numbers, while the range is only negative real numbers if x is negative and only positive real numbers if x is positive.