To solve this problem, let's analyze the pattern in the table of values given for the linear function.

Step-by-Step Solution

1. Identify the Pattern in Outputs: The table shows the following pairs for input and output:

   • Input $1$ gives Output $1$.
   • Input $2$ gives Output $-1$.
   • Input $3$ gives Output $-3$.
   • Input $4$ gives Output $-5$.

2. Find the Common Difference: Since this is a linear function, the difference between consecutive outputs should be constant.

   • From $1$ to $-1$, the difference is $-2$.
   • From $-1$ to $-3$, the difference is $-2$.
   • From $-3$ to $-5$, the difference is $-2$.

   Thus, the common difference is $-2$.

3. Determine the Linear Function Rule: With a common difference of $-2$, this is an arithmetic sequence. We can find the function by noting that the starting output (when input is 1) is 1.

   The general form of a linear function is: $f(x) = mx + b$ where $m$ is the slope (common difference) and $b$ is the y-intercept.

   Given:

   • Slope $m = -2$
   • When $x = 1$, $f(x) = 1$, so we can substitute to find $b$: $1 = (-2)(1) + b$ $1 = -2 + b$ $b = 3$

   Therefore, the function is: $f(x) = -2x + 3$

4. Find the Output When Input is $n$: Substitute $n$ into the function: $f(n) = -2n + 3$

Answer:

The output when the input is $n$ is $-2n + 3$.

Would you like more details on any part of the solution? Here are some related questions for further practice:

1. What is the output when the input is $5$ using this function?
2. How do you determine the slope of a linear function from a table of values?
3. What would the function look like if the common difference was $+3$ instead of $-2$?
4. How can you verify that a function is linear by looking at its table of values?
5. What is the y-intercept, and how does it affect the linear function?

Tip: When identifying a linear pattern, check for a constant difference between outputs to quickly determine the function's slope.