Inverse functions are functions that reverse the effect of another function. If a function f takes an input x and produces an output f(x), then its inverse function f inverse takes f(x) and returns x. This means that f of f inverse of x equals x, and f inverse of f of x also equals x.
To find the inverse of our function f(x) equals (x minus 4) over (4x plus 1), we follow a standard procedure. First, we replace f(x) with y. Then, we swap x and y variables. This gives us x equals (y minus 4) over (4y plus 1). Finally, we solve this equation for y to find our inverse function.
Starting with x equals (y minus 4) over (4y plus 1), we cross multiply to eliminate the fraction. This gives us x times (4y plus 1) equals y minus 4. Expanding the left side, we get 4xy plus x equals y minus 4. This step is crucial for simplifying rational equations.
From 4xy plus x equals y minus 4, we need to collect like terms. We move all terms containing y to one side and constants to the other side. Subtracting y from both sides gives us 4xy minus y on the left. Subtracting x from both sides gives us negative 4 minus x on the right. This organization is essential for factoring.
From 4xy minus y equals negative 4 minus x, we factor out y from the left side. This gives us y times (4x minus 1) equals negative 4 minus x. We can rewrite the right side as negative (4 plus x) to make the final step cleaner. Factoring is necessary to isolate y and solve for the inverse function.
From y times (4x minus 1) equals negative (4 plus x), we divide both sides by (4x minus 1) to solve for y. This gives us y equals negative (4 plus x) over (4x minus 1). We can rewrite this as y equals negative (x plus 4) over (4x minus 1). Therefore, our inverse function is f inverse of x equals negative (x plus 4) over (4x minus 1). Note that x cannot equal one-fourth because that would make the denominator zero.
To verify our solution, we check that f of f inverse of x equals x. We substitute our inverse function into the original function. This creates a complex fraction that we simplify step by step. After simplifying both the numerator and denominator, we find that the expression indeed equals x. A similar verification shows that f inverse of f of x also equals x, confirming our inverse is correct.
Let's analyze the domain and range of both functions. For the original function f(x) equals (x minus 4) over (4x plus 1), the denominator cannot be zero, so x cannot equal negative one-fourth. This means the domain is all real numbers except negative one-fourth. The range of f becomes the domain of f inverse, which is all real numbers except one-fourth. Similarly, the range of f inverse is all real numbers except negative one-fourth, which corresponds to the domain of f. This relationship between domains and ranges is a key property of inverse functions.