In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

Verified step by step guidance

1. Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. So, rewrite the given function as y = (x - 1)², where x ≤ 1.
2. Step 2: Swap x and y to begin solving for the inverse. This gives x = (y - 1)². Now, solve for y by taking the square root of both sides, keeping in mind the restriction x ≤ 1, which affects the sign of the square root.
3. Step 3: Solve for y by isolating it. Take the negative square root (since x ≤ 1 implies the function is decreasing) and then add 1 to both sides. This gives y = 1 - √x. Replace y with ƒ¯¹(x) to express the inverse function.
4. Step 4: To graph ƒ(x) and ƒ¯¹(x) on the same coordinate system, plot the parabola f(x) = (x - 1)² for x ≤ 1 and the inverse function ƒ¯¹(x) = 1 - √x. Remember that the graph of a function and its inverse are reflections of each other across the line y = x.
5. Step 5: Determine the domain and range of both functions. For f(x), the domain is (-∞, 1] and the range is [0, ∞). For ƒ¯¹(x), the domain is [0, ∞) and the range is (-∞, 1]. Use interval notation to express these sets.

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. For a function f(x), its inverse f¯¹(x) satisfies the condition f(f¯¹(x)) = x for all x in the domain of f¯¹. To find the inverse, one typically swaps the roles of x and y in the equation and solves for y. Understanding this concept is crucial for part (a) of the question.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input (x) and output (f(x)). For the original function f(x) and its inverse f¯¹(x), their graphs will reflect across the line y = x. This symmetry is essential for part (b) of the question, as it helps in understanding how the two functions relate to each other.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (f(x)). For the function f(x) = (x - 1)² with the restriction x ≤ 1, the domain is limited, affecting the range. Understanding how to express these sets in interval notation is necessary for part (c) of the question, as it provides a concise way to describe the function's behavior.