When studying functions, one of the most important concepts is that of the inverse function. The inverse of a function is essentially a reflection of the original function across the line y = x. It is a way of reversing the effect of the original function. So, what happens when we apply a function to its inverse? In other words, what is the value of h(f(x)) if h(x) is the inverse of f(x)?

To fully understand this concept, let us first define what it means for a function to have an inverse. A function f(x) has an inverse if it is a one-to-one function. This means that every element in the domain is mapped to a unique element in the range, and vice versa. If a function is not one-to-one, then it does not have an inverse.

To find the inverse of a one-to-one function, we swap the roles of x and y in the equation and solve for y. For example, if we have the function f(x) = 2x + 3, we can find its inverse by swapping x and y to get x = 2y + 3 and then solving for y to get y = (x – 3)/2. We then write the inverse as f^(-1)(x) = (x – 3)/2. Notice that we use the notation f^(-1)(x) to represent the inverse, instead of writing it as 1/f(x).

Now, if we have the inverse function h(x), we can apply it to any value to get its corresponding input. For example, if h(x) = 1/(x-3), then h(4) = 1/(4-3) = 1. However, what happens when we apply h(x) to the output of f(x)? In other words, what is h(f(x))?

To answer this question, we can use the definition of function composition. Function composition is when we apply one function to the output of another function. In this case, we want to apply h(x) to f(x). We write this as h(f(x)), which means we first evaluate f(x) and then plug it into h(x).

Let us use an example to illustrate this concept. Suppose we have the function f(x) = 2x + 3 and its inverse h(x) = (x – 3)/2. If we want to find the value of h(f(4)), we first need to evaluate f(4), which is 2(4) + 3 = 11. We can then plug 11 into h(x) to get h(f(4)) = h(11) = (11-3)/2 = 4.

In general, if h(x) is the inverse of f(x), then h(f(x)) = x. This is because applying the inverse undoes the effect of the original function, leaving us with the original input. To see why this is true, let us apply the definition of inverse functions. If h(x) is the inverse of f(x), then we have h(f(x)) = x for all x in the domain of f(x). This is because applying f(x) first maps x to y = f(x), and then applying h(x) to y gives us back x.

Another way to think about this is to use the graphical interpretation of inverse functions. Recall that the inverse of a function is a reflection of the original function across the line y = x. This means that the graph of f(x) and the graph of its inverse h(x) intersect at the line y = x. When we compose these two functions, we are essentially evaluating f(x) and then reflecting the output across the line y = x to get back to the original input. This is why h(f(x)) = x.

In summary, if h(x) is the inverse of f(x), then h(f(x)) = x for all x in the domain of f(x). This is because applying the inverse undoes the effect of the original function, leaving us with the original input. It is important to note that not all functions have inverses, and only one-to-one functions can have an inverse. Additionally, finding the inverse of a function can be tricky and may require solving equations or manipulating algebraic expressions. However, once we have the inverse, we can use it to undo the effects of the original function and find the original input.