The image illustrates a
mathematical function from set
A (Domain) to set
B (Codomain), defined by the rule:
$$
f(x) = x^2
$$
Let's analyze and solve this step-by-step.
---
🔹 Step 1: Understand the Components
-
Set A (Domain): The set of input values.
- Elements: {1, 2, 3}
-
Function Rule: $ f(x) = x^2 $
- This means each element in A is squared to get the output.
-
Set B (Codomain): The set of all possible output values.
- Elements: {1, 2, 3, 4, 5, 6, 7, 8, 9, 16} (as shown in the diagram)
-
Range: The actual outputs produced by the function from the domain.
- These are the values that are actually mapped to from A.
---
🔹 Step 2: Apply the Function to Each Element in Domain
We compute $ f(x) = x^2 $ for each $ x \in A $:
- $ f(1) = 1^2 = 1 $
- $ f(2) = 2^2 = 4 $
- $ f(3) = 3^2 = 9 $
So, the
outputs are: 1, 4, and 9.
---
🔹 Step 3: Identify the Range
The
range is the set of all actual outputs:
$$
\text{Range} = \{1, 4, 9\}
$$
Note: Even though the codomain includes other numbers (like 2, 3, 5, etc.), only the values that are
actually hit by the function are part of the range.
---
🔹 Step 4: Check if It’s a Valid Function
Yes, this is a valid function because:
- Every element in the domain (A) has
exactly one output in the codomain (B).
- No element in A maps to more than one value in B.
---
✔ Final Answer:
-
Domain: {1, 2, 3}
-
Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 16}
-
Range: {1, 4, 9}
-
Function Rule: $ f(x) = x^2 $
---
📌 Summary
This diagram shows how a function maps elements from the domain to the codomain using the squaring rule. Only the values 1, 4, and 9 are
actually produced, so they form the
range.
> 💡
Key Concept:
> -
Domain → inputs
> -
Codomain → all possible outputs
> -
Range → actual outputs from the given domain
✔ So, the
range of this function is
{1, 4, 9}.
Parent Tip: Review the logic above to help your child master the concept of math functions.
