Question

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)

Accepted Answer

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). 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. 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. 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. 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).

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)

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