To find the inverse equation of f(x) = 3x + 5, we first replace f(x) with y:
y = 3x + 5
Next, we switch x and y and solve for y:
x = 3y + 5
x - 5 = 3y
y = (x - 5)/3
Therefore, the inverse of f(x) is f^-1(x) = (x - 5)/3.
To find the inverse equation of g(x) = x^2 - 6, we replace g(x) with y:
y = x^2 - 6
Next, we switch x and y and solve for y:
x = y^2 - 6
x + 6 = y^2
y = ±sqrt(x + 6)
Since we want a function, we take the positive square root:
y = sqrt(x + 6)
Therefore, the inverse of g(x) is g^-1(x) = sqrt(x + 6).
The relationship between a function and its inverse is such that if (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f^-1. In other words, the roles of x and y are reversed in the inverse function. The domain of the function becomes the range of the inverse, and the range of the function becomes the domain of the inverse.
In general, not all functions have inverses. For a function to have an inverse, it must be one-to-one (i.e., each x-value in the domain maps to a unique y-value in the range). If a function is not one-to-one, then it may be possible to restrict the domain to a smaller interval so that the restricted function does have an inverse. When a function has an inverse, the domain and range of the inverse follow a pattern as stated above.