How do you work inverse functions alebraically?
Assuming y = f(x), solve for x in terms of y, and then switch the variables to get f^-1(x)
To work with inverse functions algebraically, you need to follow a specific set of steps. Here's a step-by-step explanation of how to work with inverse functions algebraically:
1. Start with the original function, let's call it f(x).
For example, let's say we have the function f(x) = 2x + 3.
2. Replace f(x) with y to represent the inverse function.
Now we have y = 2x + 3.
3. Swap the x and y variables.
Instead of y = 2x + 3, rewrite it as x = 2y + 3.
4. Solve the equation for y.
Rearrange the equation by isolating y on one side of the equation:
x - 3 = 2y
Divide both sides by 2:
(x - 3) / 2 = y
5. Replace y with f^(-1)(x) to represent the inverse function.
Now we have f^(-1)(x) = (x - 3) / 2.
So, the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x - 3) / 2.
To verify if the inverse function is correct, you can apply the composition test:
1. Start with the original function: f(x) = 2x + 3.
2. Replace f(x) with y: y = 2x + 3.
3. Replace x with f^(-1)(x): y = 2f^(-1)(x) + 3.
4. Replace y with x: x = 2f^(-1)(x) + 3.
5. Solve the equation for f^(-1)(x):
x - 3 = 2f^(-1)(x)
(x - 3) / 2 = f^(-1)(x).
If the result matches the inverse function you found earlier, then it is correct.