Did you know?
Did you know that the domain of a function refers to all possible input values, while the range represents all possible output values? In the case of the function f(x) = 3x - 5, the domain can consist of any real number, as there are no restrictions on the input. However, the range is limited to all real numbers because the function can produce any output value.
To find the inverse function, denoted as f-1(x), the roles of x and y are switched and the equation is rearranged. In this case, f-1(x) = (x + 5)/3. It is crucial to note that the domain and range of f-1(x) are swapped from those of the original function f(x). The domain of f-1(x) includes all real numbers, while its range contains any real number.
When considering (ff-1)(x) and (f-1f)(x), you may notice something interesting. Taking the composition of a function and its inverse, (ff-1)(x) and (f-1f)(x), respectively, yields the input value itself. In other words, (ff-1)(x) = x and (f-1f)(x) = x. This reveals that taking the composition of a function and its inverse undoes the effect of the function, effectively bringing you back to the original input.
Graphing the functions f(x) = 3x - 5 and f-1(x) = (x + 5)/3 can provide visualization. The graph of f will be a straight line with a slope of 3 and a y-intercept of -5. The graph of f-1 will be the reflection of f over the line y = x. When examining the graphs, you will notice that f and f-1 intersect at a point where f(x) = f-1(x), indicating a solution to the equation f = f-1.
