Mapping with the Rule x 3x - 1: Exploring the Range for the Domain 345

Introduction to Mathematical Mappings

In mathematics, a mapping (or function) is a rule that assigns to each element of one set (the domain) exactly one element of another set (the codomain). For the given problem, we are dealing with a mapping defined by the rule ( x 3x - 1 ). This rule is a transformation that can be applied to each element in the domain to determine the corresponding element in the range.

Understanding the Mapping Rule

The rule ( x 3x - 1 ) can be clarified as:

[ x 3x - 1 ]

Here, ( x ) is the input (from the domain), and the expression ( 3x - 1 ) represents the output (from the range). To apply this rule, we need to substitute each value from the domain into this expression.

Applying the Rule to the Domain 345

The domain given is 345. This means we have three individual values: 3, 4, and 5. We will apply the mapping rule ( x 3x - 1 ) to each of these values to find the corresponding range values.

Step-by-Step Solution

Let's apply the mapping rule step-by-step to each value in the domain:

Step 1: Applying the Rule to 3

[ x 3 ] Substitute ( x 3 ) into the rule ( x 3x - 1 ):

[ 3 3(3) - 1 9 - 1 8 ]

Step 2: Applying the Rule to 4

[ x 4 ]

Substitute ( x 4 ) into the rule ( x 3x - 1 ):

[ 4 3(4) - 1 12 - 1 11 ]

Step 3: Applying the Rule to 5

[ x 5 ]

Substitute ( x 5 ) into the rule ( x 3x - 1 ):

[ 5 3(5) - 1 15 - 1 14 ]

So, the range values corresponding to the domain 345 are 8, 11, and 14.

Conclusion and Additional Insights

By applying the mapping rule ( x 3x - 1 ) to each value in the domain 345, we have successfully transformed the domain values into the range values. This process is crucial in understanding the behavior of mappings and functions in mathematics.

Further Reading:

Explore more on mathematical mappings and functions: Math Is Fun - Functions and Mappings. Learn about domain and range in functions: Math Warehouse - Domain and Range. Understand more about algebraic transformations: Math Is Fun - Transformation of Functions.

If you have more questions or need help with similar problems, feel free to ask!