Final answer:
To demonstrate that f(f^{-1}(x)) = f^{-1}(f(x)) = x for a given function f(x) = -2x + 6, the inverse function f^{-1}(x) is found and substitution is performed, confirming that each composition of function and its inverse results in x.
Step-by-step explanation:
To show that f(f^{-1}(x)) = f^{-1}(f(x)) = x for the function f(x) = -2x + 6, we first need to find the inverse function f^{-1}(x). The inverse function 'undoes' what the original function does. Following the steps of algebraically finding an inverse function, we set y = -2x + 6 and then solve for x:
- y = -2x + 6
- y - 6 = -2x
- (y - 6)/-2 = x
So the inverse function is f^{-1}(x) = (x - 6)/-2.
Next, we substitute f^{-1}(x) into f(x):
- f(f^{-1}(x)) = f((x - 6)/-2)
- = -2((x - 6)/-2) + 6
- = x - 6 + 6
- = x
We see that f(f^{-1}(x)) indeed equals x. Now, we plug f(x) into f^{-1}(x):
- f^{-1}(f(x)) = f^{-1}(-2x + 6)
- = ((-2x + 6) - 6)/-2
- = -2x/-2
- = x
Thus, f^{-1}(f(x)) also equals x, confirming the original statement that f(f^{-1}(x)) = f^{-1}(f(x)) = x.