I am too embarassed to ask this Calculus (really pre-calculus) question in tutoring, because I know I should know. Is the inverse of f(x)=3x-1 actually f(x)=1/3x+1? How do I find it? What if it asks the same equation replaced with f to the -1 power (x)? I think I know how the F of -1 power (x) is x+1/3, but I know that from watching someone else.
Ok. An inverse function is formed from switch in the input and output variables(x,y).
so y = 3x -1
flip x = 3y-1 and solve for y again
x+1=3y
y = 1/3(x+1)
so f^-1(x) = 1/3(x+1)
and the inverse of an inverse function is the original equation: f(x) = 3x-1
okay, thanks!
No need to be embarrassed! Understanding concepts in math can sometimes be challenging, but it's always better to ask questions and seek clarification. Let's go through finding the inverse of a function step by step.
To find the inverse of a function, you can follow these steps:
1. Start with the original function, f(x). In this case, f(x) = 3x - 1.
2. Replace f(x) with y. Then, swap x and y to interchange their roles. The equation becomes x = 3y - 1.
3. Rearrange the equation to solve for y. Add 1 to both sides to isolate the term 3y, resulting in x + 1 = 3y.
4. Divide both sides by 3 to solve for y. The equation becomes (x + 1)/3 = y.
5. Replace y with f^(-1)(x) to denote the inverse function. So, f^(-1)(x) = (x + 1)/3.
Therefore, the inverse function of f(x) = 3x - 1 is f^(-1)(x) = (x + 1)/3.
If you encounter the notation f^(-1)(x), it represents the inverse function of f(x). The process of finding the inverse remains the same; we want to swap x and y and solve for y.
Let's apply the same steps to find f^(-1)(x) for the given equation f(x) = 1/3x + 1:
1. Start with f(x) = 1/3x + 1.
2. Replace f(x) with y. The equation becomes y = 1/3x + 1.
3. Swap x and y: x = 1/3y + 1.
4. Isolate y on one side by subtracting 1 from both sides: x - 1 = 1/3y.
5. Multiply both sides by 3: 3(x - 1) = y.
6. Simplify: 3x - 3 = y.
So, the inverse of f(x) = 1/3x + 1 is f^(-1)(x) = 3x - 3.
Remember, it's essential to validate your answers by checking that the composition of a function and its inverse yields the original input value.
I hope this explanation helps! If you have any further questions, feel free to ask.