The function f(x)=2x-3 is one-to-one. Its inverse function is f^{-1}(x)=(x+3)/2. The domain and range of both f and f^{-1} are all real numbers.
To determine whether f(x)=2x-3 is one-to-one, we can use the following definition:
A function is one-to-one if and only if no two different inputs produce the same output.
In other words, for any two distinct inputs x_1 and x_2, we must have f(x_1) != f(x_2).
Let's assume that f(x_1) = f(x_2) for some distinct inputs x_1 and x_2. Then we have:
2x_1 - 3 = 2x_2 - 3
Subtracting 3 from both sides, we get:
2x_1 = 2x_2
Dividing both sides by 2, we get:
x_1 = x_2
This contradiction shows that our original assumption must have been wrong, and therefore f(x)=2x-3 is one-to-one.
Part(a) (To find the inverse function of f(x)=2x-3, we need to swap the roles of x and y. Solving for y in the equation y=2x-3, we get:
y = 2x - 3
y + 3 = 2x
x = (y + 3)/2
Therefore, the inverse function of f(x)=2x-3 is:
f^{-1}(x) = (x + 3)/2
Part (b)
The following graph shows f(x)=2x-3 and its inverse function f^{-1}(x)=(x+3)/2 on the same axes:
graph(y = 2x - 3, color="blue")
graph(y = (x + 3)/2, color="red")
Part (c)
The domain of f(x)=2x-3 is all real numbers. The range of f(x)=2x-3 is also all real numbers.
The domain of f^{-1}(x)=(x+3)/2 is all real numbers. The range of f^{-1}(x)=(x+3)/2 is also all real numbers.