Final answer:
The given relation is a function but not one-to-one. The inverse of the function is y^-1 = sqrt((x + 8)/2) + 4. The function and its inverse are proven to be inverses algebraically.
Step-by-step explanation:
Part A: To determine if the given relation is a function, we need to check if each input value (x) has only one output value (y).
To do this, we can use the vertical line test. If a vertical line intersects the graph at more than one point, then the relation is not a function.
For the given equation, y = 2(x - 4)^2 - 8, when we plot it on a graph, we find that every input value (x) has only one output value (y), and no vertical line intersects the graph at more than one point. Therefore, the given relation is a function.
To determine if the function is one-to-one, we need to check if each input value (x) has a unique output value (y). One way to test this is to perform the horizontal line test. If a horizontal line intersects the graph at more than one point, then the function is not one-to-one.
For the given equation, when we plot it on a graph, we find that a horizontal line can intersect the graph at multiple points. Therefore, the given relation is not one-to-one.
To make the relation one-to-one, we can restrict the domain by considering a specific range of x-values. For example, we can restrict x to only positive values (x > 0) or x to only negative values (x < 0). This will result in a function that has a unique output value (y) for each input value (x), thus making it one-to-one.
Part B: To find the inverse of the given function, we need to solve the equation for x in terms of y.
Starting with y = 2(x - 4)^2 - 8, we can rearrange the equation to solve for x:
y + 8 = 2(x - 4)^2
(y + 8)/2 = (x - 4)^2
sqrt((y + 8)/2) = x - 4
sqrt((y + 8)/2) + 4 = x
Now, we replace x with y and y with x to find the inverse function:
x = sqrt((y + 8)/2) + 4
y = sqrt((x + 8)/2) + 4
This is the inverse function, written as y^-1. However, we can simplify it further:
y^-1 = sqrt((x + 8)/2) + 4
Part C: To prove that y and y^-1 are inverse functions, we need to show that applying one function after the other results in the original input value.
If we substitute y^-1 into the original function y = 2(x - 4)^2 - 8:
y = 2((sqrt((x + 8)/2) + 4) - 4)^2 - 8
y = 2(sqrt((x + 8)/2))^2 - 8
y = 2((x + 8)/2) - 8
y = (x + 8) - 8
y = x
This proves that y and y^-1 are inverse functions, as applying y^-1 to y or y to y^-1 results in the original input value.