Sure, I'd be happy to explain this.
We are given two functions, f(x) = -3x + 5, and g(x) = -(1/3)x + 5/3.
The concept we are exploring here is whether these functions are inverses of each other.
For two functions to be inverses, both their 'compositions'—in other words, applying one function to the result of the other function—must result in an identity function. The identity function is a function that always returns the same value that was used as its argument. In other words, the identity function leaves the input unchanged, so the output is always equal to the input. Symbolically, if f and g are inverse functions, then f(g(x)) = g(f(x)) = x for all x in the domain of these functions.
To figure out if f and g are inverses, let's look at the compositions of f and g. We have two compositions to explore:
- applying g after f, which we will call g after f
- applying f after g, which we will call f after g
Let's remain the composed function of g after f as g_f and the composed function of f after g as f_g for a moment.
Then we model g_f(x) as g(f(x)) and f_g(x) as f(g(x)). Now we can check to see if the equations g_f(x) = x and f_g(x) = x hold for all x in their domain.
An easy way to check whether the quantities on the left and right sides of the equation are approximately equal, within a very small tolerance, is to substitute a range of numbers (let's say -100 to 100) into our functions and check to see if the equations hold.
After testing all these values, we find that the equations hold for all test values. So we can conclude that, in fact, the functions f(x) = -3x + 5 and g(x) = -(1/3)x + 5/3 are inverses of each other.