Questions: Refer to functions n, p. Evaluate the function and Write the domain in interval notation. n(x)=x-6 p(x)=x^2+5 x Part: 0 / 4 Part 1 of 4 (n circ p)(x)= (Choose one) nabla

Refer to functions n, p. Evaluate the function and Write the domain in interval notation.

n(x)=x-6 p(x)=x^2+5 x

Part: 0 / 4

Part 1 of 4

(n circ p)(x)= (Choose one) nabla

Transcript text: Refer to functions $n, p$. Evaluate the function and Write the domain in interval notation. \[ n(x)=x-6 \quad p(x)=x^{2}+5 x \] Part: $0 / 4$ Part 1 of 4 \[ (n \circ p)(x)=\text { (Choose one) } \nabla \]

Solution

Solution Steps

To evaluate the composition of functions $ (n \circ p)(x) $, we need to substitute the function $ p(x) $ into the function $ n(x) $. This means replacing every instance of $ x $ in $ n(x) $ with $ p(x) $. After finding the expression for $ (n \circ p)(x) $, we will determine the domain of the resulting function. The domain of a polynomial function is all real numbers, so we will express this in interval notation.

Step 1: Evaluate the Composition of Functions

To find $ (n \circ p)(x) $, we substitute $ p(x) = x^2 + 5x $ into $ n(x) = x - 6 $. This gives us: \[ (n \circ p)(x) = p(x) - 6 = (x^2 + 5x) - 6 = x^2 + 5x - 6 \]

Step 2: Determine the Domain

The function $ x^2 + 5x - 6 $ is a polynomial. The domain of any polynomial function is all real numbers. Therefore, we express the domain in interval notation as: \[ \text{Domain} = (-\infty, \infty) \]