An inverse function reverses the effect of the original function. If we have a function f that maps x to y, then its inverse function f-inverse maps y back to x. In this lesson, we'll find the inverse of f of x equals (x minus 3) divided by (4x plus 1).
To find the inverse of a function, we start by replacing f of x with y. Then we swap x and y in the equation. So if y equals (x minus 3) divided by (4x plus 1), we rewrite this as x equals (y minus 3) divided by (4y plus 1). Our goal is to solve this new equation for y, which will give us the inverse function.
Now we need to solve for y. First, we eliminate the fraction by multiplying both sides by the denominator (4y plus 1). This gives us x times (4y plus 1) equals y minus 3. Expanding the left side, we get 4xy plus x equals y minus 3. We've now eliminated the fraction and have a linear equation in terms of x and y.
Next, we collect all terms containing y on one side of the equation. Moving y to the left side and x to the right side, we get 4xy minus y equals negative 3 minus x. Now we can factor out y from the left side, giving us y times (4x minus 1) equals negative 3 minus x. This step prepares us to isolate y completely.
To solve for y, we divide both sides by the coefficient of y, which is (4x minus 1). This gives us y equals (negative 3 minus x) divided by (4x minus 1). We can factor out a negative sign from the numerator to get y equals negative (3 plus x) divided by (4x minus 1), which simplifies to y equals (negative x minus 3) divided by (4x minus 1).
Since we've solved for y in terms of x, we can now write our inverse function. Replacing y with f-inverse of x, we get f-inverse of x equals (negative x minus 3) divided by (4x minus 1). Note that this function has a domain restriction: x cannot equal one-fourth, because that would make the denominator zero. This corresponds to a vertical asymptote in the graph of the inverse function.
To verify our answer, we need to check that f of f-inverse of x equals x. We substitute our inverse function into the original function. This gives us a complex fraction which we need to simplify. After combining terms in the numerator and denominator, we'll see if the complex fraction simplifies to just x, confirming our inverse function is correct.
Let's finish the verification. Simplifying the numerator: negative x minus 3 minus 12x plus 3 equals negative 13x. Simplifying the denominator: negative 4x minus 12 plus 4x minus 1 equals negative 13. So we have (negative 13x over (4x minus 1)) divided by (negative 13 over (4x minus 1)). Dividing by a fraction is the same as multiplying by its reciprocal, giving us (negative 13x over (4x minus 1)) times ((4x minus 1) over negative 13). The (4x minus 1) terms cancel out, as do the negative 13 terms, leaving us with just x. This confirms our inverse function is correct. The final answer is f-inverse of x equals (negative x minus 3) divided by (4x minus 1), where x cannot equal one-fourth.