Question Statement

Check the type of function $g(x)=2x^2+3x+1$ given the following:

• Domain of $g$: $\{0,1,2,3\}$
• Codomain of $g$: $\{1,6,15,28,35\}$

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Background and Explanation

To determine the type of function, we evaluate the function for every element in its domain. A function is one-to-one (injective) if every input produces a unique output, and it is an into function if the range (the set of actual outputs) is a proper subset of the codomain.

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Solution

To identify the type of function, we first calculate the value of $g(x)$ for each element in the domain $\{0,1,2,3\}$.

Step 1: Evaluate the function for each element in the domain

For $x=0$: $g(0)=2(0{)}^2+3(0)+1=0+0+1=1$

For $x=1$: $g(1)=2(1{)}^2+3(1)+1=2+3+1=6$

For $x=2$: $g(2)=2(2{)}^2+3(2)+1=2(4)+6+1=8+6+1=15$

For $x=3$: $g(3)=2(3{)}^2+3(3)+1=2(9)+9+1=18+9+1=28$

Step 2: Check for One-to-One (Injective) property

We observe the outputs: $g(0)=1,g(1)=6,g(2)=15,g(3)=28$

Since all the outputs are distinct ($g(0)\ne g(1)\ne g(2)\ne g(3)$), every element in the domain maps to a unique element in the codomain. Therefore, the function is one-to-one (1-1).

Step 3: Check for Into/Onto property

The Range of the function is the set of actual outputs: $\text{Range }g=\{1,6,15,28\}$

The given Codomain is: $\text{Codomain }g=\{1,6,15,28,35\}$

Since the Range is a proper subset of the Codomain ($\text{Range }g\subset \text{Codomain }g$)—specifically, the element $35$ is in the codomain but is not mapped to by any element in the domain—the function is an into function.

Conclusion: The function $g(x)$ is a one-to-one and into function.

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Key Formulas or Methods Used

• Function Evaluation: Substituting domain values into $g(x)=a{x}^2+bx+c$.
• One-to-One (Injective): A function where $f(x_1)=f(x_2)$ implies $x_1=x_2$.
• Into Function: A function where the Range is not equal to the Codomain ($\text{Range}\subset \text{Codomain}$).

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Summary of Steps

1. Substitute each value from the domain $\{0,1,2,3\}$ into the function $g(x)=2x^2+3x+1$.
2. List the resulting values to find the Range: $\{1,6,15,28\}$.
3. Compare the outputs to see if they are unique; since they are, the function is one-to-one.
4. Compare the Range to the provided Codomain; since the Range is a subset and not equal to the Codomain, the function is into.