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. ƒ(x) = x² − 4, x ≥ 0

Accepted Answer

Step 1: To find the inverse function ƒ¯¹(x), start by replacing ƒ(x) with y. So, rewrite the given function as y = x² - 4, where x ≥ 0. Then, swap x and y to begin solving for y. This gives x = y² - 4. Step 2: Solve for y in terms of x. Add 4 to both sides of the equation to isolate the y² term: x + 4 = y². Then, take the square root of both sides to solve for y. Since x ≥ 0, we only consider the positive square root: y = √(x + 4). Thus, the inverse function is ƒ¯¹(x) = √(x + 4). Step 3: To graph ƒ(x) and ƒ¯¹(x) on the same coordinate system, plot the parabola ƒ(x) = x² - 4 for x ≥ 0 (this is the right half of the parabola). Then, plot the square root function ƒ¯¹(x) = √(x + 4), which is the reflection of ƒ(x) across the line y = x. Step 4: Determine the domain and range of ƒ(x). Since ƒ(x) = x² - 4 and x ≥ 0, the domain of ƒ(x) is [0, ∞). The range of ƒ(x) is [-4, ∞) because the smallest value of ƒ(x) occurs when x = 0, giving ƒ(0) = -4. Step 5: Determine the domain and range of ƒ¯¹(x). The domain of ƒ¯¹(x) is the range of ƒ(x), which is [-4, ∞). The range of ƒ¯¹(x) is the domain of ƒ(x), which is [0, ∞). Use interval notation to express these relationships clearly.

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. ƒ(x) = x² − 4, x ≥ 0

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inverse Functions

Domain and Range

Graphing Functions