Choose any two specific functions that have inverses. Use your chosen functions to answer any one of the following questions
- If the inverses of two functions are both functions, will the inverse of the sum or difference of the original functions also be a function?
Can someone help me?
first off, take a little review of what defines a function.
You will note that all these relations are 1-to-1.
Now, does adding or subtracting change that?
Of course, I'd be happy to help you with your question! To answer the question about the inverse of the sum or difference of functions, we need to select two specific functions that have inverses.
Let's choose the functions f(x) = 2x and g(x) = x/4. Both of these functions have inverses. To find their inverses, we need to swap the roles of x and y and solve for y.
For f(x) = 2x, we swap x and y to get x = 2y and solve for y: y = x/2. So, the inverse of f(x) is f^(-1)(x) = x/2.
For g(x) = x/4, swapping x and y gives x = y/4. Solving for y, y = 4x. Therefore, the inverse of g(x) is g^(-1)(x) = 4x.
Now let's consider the sum and difference of these functions.
The sum of f(x) and g(x) is h(x) = f(x) + g(x) = 2x + x/4. To find the inverse of h(x), we need to find the inverse of f(x) + g(x).
Using the derived inverses, f^(-1)(x) = x/2 and g^(-1)(x) = 4x, we can rewrite h(x) as h(x) = f^(-1)(x) + g^(-1)(x) = x/2 + 4x.
To find the inverse of h(x), we need to swap x and y and solve for y. Let's call the inverse of h(x) as h^(-1)(x).
Swapping x and y: x = y/2 + 4y.
To solve for y, we need to isolate y. Let's multiply both sides of the equation by 2 to get rid of the fraction: 2x = y + 8y.
Combining like terms, we get 2x = 9y. Dividing both sides by 9: y = (2/9)x.
Therefore, the inverse of h(x) = x/2 + 4x is h^(-1)(x) = (2/9)x. As we can see, the inverse of the sum of the original functions is a function.
Similarly, you can also find the inverse of the difference of the original functions if desired.