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? Explain your reasoning.
what if
y = x^3 ----> y =x^1/3 is function
y = x ---> y = x is function
2 y = x^3 + x ---> y+y^3 = 2x
y(1+y^2) =2 x
for x = 0
y = 0 or y = +/-i
more than one value off y for x = 0, no, no for function
To determine whether the inverse of the sum or difference of two functions will also be a function when the inverses of the original functions are both functions, we need to consider the definition and properties of inverse functions.
Let's assume we have two functions, f(x) and g(x), with their inverses denoted as f^(-1)(x) and g^(-1)(x), respectively.
1. Inverse of the sum:
If we want to find the inverse of the sum, h(x) = f(x) + g(x), we need to determine whether h(x) has an inverse that is also a function.
To find the inverse of h(x), we apply the process of finding the inverse to h(x) = f(x) + g(x):
- Write h(x) in terms of x: h(x) = f(x) + g(x)
- Swap x and y: x = f^(-1)(y) + g^(-1)(y)
- Solve for y: y = (x - g^(-1)(y))/f^(-1)(y)
If f^(-1)(y) and g^(-1)(y) are both functions, then their respective domains must be such that we do not get a division by zero or any other undefined operation. Additionally, the denominator, f^(-1)(y), cannot be equal to zero, as division by zero is not defined.
Therefore, if the inverses of f(x) and g(x) are both functions, and their domains satisfy these conditions, then the inverse of the sum, h^(-1)(x), will also be a function.
2. Inverse of the difference:
Similarly, if we want to find the inverse of the difference, h(x) = f(x) - g(x), we follow the same process as above:
- Write h(x) in terms of x: h(x) = f(x) - g(x)
- Swap x and y: x = f^(-1)(y) - g^(-1)(y)
- Solve for y: y = (x + g^(-1)(y))/f^(-1)(y)
Again, if f^(-1)(y) and g^(-1)(y) are both functions, and their domains satisfy the conditions to avoid division by zero or other undefined operations, then the inverse of the difference, h^(-1)(x), will also be a function.
In conclusion, if the inverses of two functions are both functions, the inverses of their sum and difference will also be functions, given that the domains are appropriately defined.