a. The function 𝑓(𝑥) = 𝑥² + 2 does not have an inverse for all values of 𝑥. This is because the function is not one-to-one, meaning that there are multiple values of 𝑥 that can produce the same output.

b. The largest possible domain where 𝑓(𝑥) has an inverse is when 𝑥 ≥ 0. This is because for 𝑥 ≥ 0, the function 𝑓(𝑥) is one-to-one and therefore has an inverse. The inverse function 𝑓−1(𝑥) can be found by swapping 𝑥 and 𝑦 in the original equation and solving for 𝑦:

𝑥 = 𝑦² + 2
𝑦² = 𝑥 - 2
𝑦 = sqrt(𝑥 - 2)

So, 𝑓−1(𝑥) = sqrt(𝑥 - 2).

c. The domain of 𝑓−1(𝑥) is 𝑥 ≥ 2, because the square root of a negative number is not a real number. The range of 𝑓−1(𝑥) is 𝑦 ≥ 0, because the square root of a number is always nonnegative.

Question

a. The function 𝑓(𝑥) = 𝑥² + 2 does not have an inverse for all values of 𝑥. This is because the function is not one-to-one, meaning that there are multiple values of 𝑥 that can produce the same output.

b. The largest possible domain where 𝑓(𝑥) has an inverse is when 𝑥 ≥ 0. This is because for 𝑥 ≥ 0, the function 𝑓(𝑥) is one-to-one and therefore has an inverse. The inverse function 𝑓−1(𝑥) can be found by swapping 𝑥 and 𝑦 in the original equation and solving for 𝑦:

𝑥 = 𝑦² + 2
𝑦² = 𝑥 - 2
𝑦 = sqrt(𝑥 - 2)

So, 𝑓−1(𝑥) = sqrt(𝑥 - 2).

c. The domain of 𝑓−1(𝑥) is 𝑥 ≥ 2, because the square root of a negative number is not a real number. The range of 𝑓−1(𝑥) is 𝑦 ≥ 0, because the square root of a number is always nonnegative.

Knowee AI · Accepted Answer

a. The function 𝑓(𝑥) = 𝑥² + 2 does not have an inverse for all values of 𝑥. This is because the function is not one-to-one, meaning that there are multiple values of 𝑥 that can produce the same output.

b. The largest possible domain where 𝑓(𝑥) has an inverse is when 𝑥 ≥ 0. This is because for 𝑥 ≥ 0, the function 𝑓(𝑥) is one-to-one and therefore has an inverse. The inverse function 𝑓−1(𝑥) can be found by swapping 𝑥 and 𝑦 in the original equation and solving for 𝑦:

𝑥 = 𝑦² + 2
𝑦² = 𝑥 - 2
𝑦 = sqrt(𝑥 - 2)

So, 𝑓−1(𝑥) = sqrt(𝑥 - 2).

c. The domain of 𝑓−1(𝑥) is 𝑥 ≥ 2, because the square root of a negative number is not a real number. The range of 𝑓−1(𝑥) is 𝑦 ≥ 0, because the square root of a number is always nonnegative.

Given a function: 𝑓(𝑥) = 𝑥2 + 2.a. Does 𝑓(𝑥) has an inverse for all values of 𝑥.b. Find a possible largest domain where 𝑓(𝑥) has an inverse. Then find the inversefunction 𝑓−1(𝑥).c. What is the domain and range of 𝑓−1(𝑥)

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