State the inverse of the following function, AND state the Domain and Range of the inverse. f(x)=x-2
f(x) = x-2
f^-1(x) = x+2
check to be sure f(f^-1(x)) = f^-1(f(x)) = x
As with all polynomials, the domain is all real numbers.
The line is not horizontal, so it extends up and down forever. so the range is also all reals.
To find the inverse of the function f(x) = x - 2, we need to swap the x and y variables and solve for y.
Step 1: Replace f(x) with y.
y = x - 2
Step 2: Swap x and y.
x = y - 2
Step 3: Solve for y.
Adding 2 to both sides: x + 2 = y
Rearranging: y = x + 2
So, the inverse of the function f(x) = x - 2 is f^(-1)(x) = x + 2.
Now let's determine the domain and range of the inverse function.
The domain of the original function f(x) = x - 2 is all real numbers (-∞, ∞).
The range of the original function is also all real numbers (-∞, ∞), as any value of x can be substituted to get a corresponding y value.
When we find the inverse function f^(-1)(x) = x + 2, the domain and range swap places.
So, the domain of the inverse function f^(-1)(x) is all real numbers (-∞, ∞).
And the range of the inverse function f^(-1)(x) is also all real numbers (-∞, ∞).