Final Answer:
The function that represents the inverse of f(x) = 2x + 3 is B) f⁻¹(x) = 1/2 x - 3/2.
Step-by-step explanation:
The inverse function, denoted as f⁻¹(x), essentially "undoes" the operations of the original function. For the given function f(x) = 2x + 3, to find its inverse, we need to swap the roles of x and y and solve for y. Let y = 2x + 3. Swap x and y: x = 2y + 3. Now, solve for y:
- Subtract 3 from both sides: x - 3 = 2y.
- Divide both sides by 2: (x - 3)/2 = y.
- Therefore, f⁻¹(x) = (x - 3)/2.
Now, we can simplify this expression to match one of the given options. Distribute the 1/2:
1. f⁻¹(x) = (1/2)x - 3/2.
Thus, the correct answer is B) f⁻¹(x) = 1/2 x - 3/2. This result makes sense as it reflects the reversal of the operations performed by the original function.
In summary, the inverse function undoes the operations of the original function. The calculations involve swapping x and y, solving for y, and simplifying the expression. The correct answer, B), aligns with these steps and represents the inverse function of f(x) = 2x + 3 as f⁻¹(x) = 1/2 x - 3/2.