Question

to which family does the function \\(y = (x+2)^{\frac{1}{2}} + 3\\) belong?

• quadratic
• square root
• exponential
• reciprocal

Explanation:

🆕 New Concept Discovered: Parent Functions and Families
Identifying the core shape and equation of a function.

Step 1: Analyze the exponent

The given function is:

$$ y = (x+2)^{\frac{1}{2}} + 3 $$

We look at the exponent applied to the variable term, which is \(\frac{1}{2}\).

Step 2: Rewrite using radical notation

An exponent of \(\frac{1}{2}\) is mathematically equivalent to taking the square root of the base:

$$ (x+2)^{\frac{1}{2}} = \sqrt{x+2} $$

Substituting this back into the equation gives:

$$ y = \sqrt{x+2} + 3 $$

Step 3: Identify the function family

The parent function of this equation is \( y = \sqrt{x} \). Therefore, this function belongs to the square root family.

Answer:

square root