Questions: Consider the following function. u(x)=-3 ∛(x-3)+3 Step 2 of 2: Determine the domain and range of the original function. Express your answer in interval notation.

Consider the following function.
u(x)=-3 ∛(x-3)+3

Step 2 of 2: Determine the domain and range of the original function. Express your answer in interval notation.

Transcript text: Consider the following function. \[ u(x)=-3 \sqrt[3]{x-3}+3 \] Step 2 of 2: Determine the domain and range of the original function. Express your answer in interval notation.

Solution

Solution Steps

Step 1: Find the domain of u(x)

The function u(x) involves a square root, and the expression inside the square root must be greater than or equal to zero. So, we set up the inequality:

x - 3 >= 0

Adding 3 to both sides, we get:

x >= 3

The domain of u(x) is all x values greater than or equal to 3. In interval notation, this is [3, ∞).

Step 2: Find the range of u(x)

The square root of any number is always non-negative (zero or positive). Thus, √(x-3) will be greater than or equal to 0. Multiplying this by -3 yields a value less than or equal to 0. Adding 3 shifts this up by 3. The range of -3√(x-3) would be (-∞, 0]. Adding 3 to every output, shifts the range to (-∞, 3].

Final Answer:

Domain: [3, ∞)

Range: (-∞, 3]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!