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)

Verified step by step guidance

1. Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = √(x - 1). Next, swap x and y to reflect the inverse relationship. This results in x = √(y - 1).
2. Step 2: Solve for y in terms of x to find the inverse function. Square both sides of the equation to eliminate the square root: x² = y - 1. Then, isolate y by adding 1 to both sides: y = x² + 1. Thus, the inverse function is ƒ¯¹(x) = x² + 1.
3. Step 3: To graph both f(x) and ƒ¯¹(x), plot f(x) = √(x - 1), which is a square root function starting at x = 1 and increasing to the right. For ƒ¯¹(x) = x² + 1, plot a parabola that opens upwards with its vertex at (0, 1). Ensure both graphs are in the same rectangular coordinate system.
4. Step 4: Determine the domain and range of f(x). Since f(x) = √(x - 1), the domain is [1, ∞) because the square root function is only defined for x - 1 ≥ 0. The range is [0, ∞) because the square root function outputs non-negative values.
5. Step 5: Determine the domain and range of ƒ¯¹(x). Since ƒ¯¹(x) = x² + 1, the domain is [0, ∞) because the input to the inverse function corresponds to the range of f(x). The range of ƒ¯¹(x) is [1, ∞) because the output of the inverse function corresponds to the domain of f(x).

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.

Domain and Range

The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. Understanding the domain and range is crucial for both the original function and its inverse, as they are interchanged in inverse functions.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input and output values. When graphing both a function and its inverse, the graphs will be symmetric with respect to the line y = x, illustrating how each function undoes the other.