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 of f and ƒ¯¹. f(x)=2x-1

Accepted Answer

Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. The given function is f(x) = 2x - 1, so rewrite it as y = 2x - 1. Step 2: Swap x and y in the equation to begin solving for the inverse. This gives x = 2y - 1. Step 3: Solve for y in terms of x. Add 1 to both sides to isolate the term with y: x + 1 = 2y. Then divide both sides by 2 to solve for y: y = (x + 1)/2. The inverse function is ƒ¯¹(x) = (x + 1)/2. Step 4: To graph ƒ and ƒ¯¹(x) on the same rectangular coordinate system, plot the original function f(x) = 2x - 1 (a straight line with slope 2 and y-intercept -1) and the inverse function ƒ¯¹(x) = (x + 1)/2 (a straight line with slope 1/2 and y-intercept 1/2). Note that the graphs of f and ƒ¯¹(x) are reflections of each other across the line y = x. Step 5: Use interval notation to describe the domain and range. For f(x) = 2x - 1, the domain is all real numbers (-∞, ∞) and the range is also all real numbers (-∞, ∞). For ƒ¯¹(x) = (x + 1)/2, the domain is all real numbers (-∞, ∞) and the range is all real numbers (-∞, ∞).

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 of f and ƒ¯¹. f(x)=2x-1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inverse Functions

Graphing Functions

Domain and Range