Answer:
Step-by-step explanation:
To find the value of m, we need to determine the condition under which the inverse of the function f(x) = (x - 3)^2 + 1 exists.
The inverse of a function f(x) can be found by swapping the x and y variables and solving for y.
1. Swap x and y: x = (y - 3)^2 + 1. 2. Solve for y: x - 1 = (y - 3)^2. 3. Take the square root of both sides: √(x - 1) = y - 3. 4. Add 3 to both sides: y = √(x - 1) +3. The inverse function of f(x) is given by f^(-1)(x) = √(x - 1) + 3.
Now, let's determine the condition under which the inverse function exists. The square root function (√) is defined for non-negative real numbers. Therefore, the expression (x - 1) inside the square root must be greater than or equal to 0. x - 1 ≥ 0 x ≥ 1 So, the inverse function exists for x ≥ 1. Therefore, the value of m is 1 or any number greater than 1. In summary, for the function f(x) = (x - 3)^2 + 1, the inverse function exists for x ≥ 1.
Thus, m can take any value greater than or equal to 1
