Question

What is the inverse of the function f(x)=3(x−5)² −4, given that x≤−5? (2 points)

Answer 1

The inverse of the function f(x) = 3(x−5)² −4, given that x≤−5, is found by solving for x in terms of y and then swapping x and y. The steps include isolating the squared term, extracting the square root, and then solving for x. The inverse is f⁻¹(x) = √((x + 4) / 3) + 5.

Step-by-step explanation:

The inverse of a function 'f(x)' is found by following some standard algebraic steps that essentially reverse the original function's operations. For the function f(x) = 3(x−5)² −4, the goal is to express 'x' in terms of 'y' (which will be the new 'x' in the inverse function, typically denoted as 'f⁻¹(x)'). Here’s a step-by-step solution:

1. Let y = f(x). So, we substitute 'f(x)' with 'y': y = 3(x−5)² −4.
2. Add 4 to both sides: y + 4 = 3(x−5)².
3. Divide both sides by 3: (y + 4) / 3 = (x−5)².
4. Take the square root of both sides. Because we are given that x≤−5, we use the negative square root: x − 5 = √((y + 4) / 3).
5. Lastly, we solve for 'x': x = √((y + 4) / 3) + 5.

The inverse function, therefore, given that x≤−5, is f⁻¹(x) = √((x + 4) / 3) + 5.