Determine whether the given function is one-to-one. If it is one-to-one, find its inverse function.
-f(x) = 6x + 6
D) not a one-to-one function

Question

Quizplus · Accepted Answer

To check whether the function is one-to-one, we need to verify that each element in the domain corresponds to a unique element in the range. 
Let f(x1) = f(x2), where x1 and x2 are different values in the domain of f. 
Then, 6x1 + 6 = 6x2 + 6 
Simplifying this equation, we get: 
6x1 = 6x2 
x1 = x2 
Thus, we have shown that if f(x1) = f(x2), then x1 = x2. 
Therefore, the function is one-to-one. 
To find the inverse of f(x) = 6x + 6, we need to interchange the roles of x and f(x) and solve for f(x): 
y = 6x + 6 
x = 6y + 6 
x - 6 = 6y 
y = (x - 6)/6 
Therefore, the inverse function of f(x) is f⁻¹(x) = (x - 6)/6. To test if a function is one-to-one, we can use the horizontal line test. If every horizontal line intersects the graph of the function at most once, then the function is one-to-one. In this case, the graph of f(x) = 6x + 6 is a straight line with a slope of 6, which means that it increases without bound as x moves to the right and decreases without bound as x moves to the left. Therefore, no horizontal line can intersect the graph of f(x) more than once. So f(x) is a one-to-one function. To find its inverse function, we can solve for x in terms of f(x):

f(x) = 6x + 6
f(x) - 6 = 6x
x = (f(x) - 6)/6

So the inverse function is g(x) = (x - 6)/6, or equivalently, g(f(x)) = (6x + 6 - 6)/6 = x/6 + 1.

Determine Whether the Given Function Is One-To-One