Let $\mathbf{N}$ be the set of natural numbers. Define a real valued function $f: \mathbf{N} \rightarrow \mathbf{N}$ by $f(x)=2 x+1$. Using this definition, complete the table given below. $$\begin{array}{|r|c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & f(1)=\ldots & f(2)=\ldots & f(3)=\ldots & f(4)=\ldots & f(5)=\ldots & f(6)=\ldots & f(7)=\ldots \\ \hline \end{array}$$

Step 1: Understand the Function Definition

The problem defines a function $$f(x) = 2x + 1$$ . This means that for any input value $$x$$ , we multiply $$x$$ by 2 and then add 1 to get the output value f(x). The domain and codomain are the set of natural numbers $$\mathbf{N}$$ .

Step 2: Calculate f(1)

To find $$f(1)$$ , we substitute $$x=1$$ into the function definition. This gives us $$2 \times 1 + 1$$ .

Step 3: Calculate f(2)

Next, we find $$f(2)$$ by substituting $$x=2$$ into the function. This results in $$2 \times 2 + 1$$ .

Step 4: Calculate f(3)

For $$f(3)$$ , we substitute $$x=3$$ into the function, yielding $$2 \times 3 + 1$$ .

Step 5: Calculate f(4), f(5), f(6), f(7)

We continue the process for the remaining values of $$x$$ . For $$x=4$$ , $$f(4) = 2(4)+1 = 9$$ . For $$x=5$$ , $$f(5) = 2(5)+1 = 11$$ . For $$x=6$$ , $$f(6) = 2(6)+1 = 13$$ . For $$x=7$$ , $$f(7) = 2(7)+1 = 15$$ .

Step 6: Complete the Table

By calculating the function values for each given $$x$$ , we can now complete the table.

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