To find the output when the input is 6 given the input-output pairs of a linear function, follow these steps:
1. Identify the given input-output pairs and express them as coordinates:
- (1, 3)
- (2, 9)
- (3, 15)
2. Calculate the slope (m) of the linear function:
- The slope [tex]\(m\)[/tex] is calculated using two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] with the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Using the points (1, 3) and (2, 9):
[tex]\[
m = \frac{9 - 3}{2 - 1} = \frac{6}{1} = 6
\][/tex]
3. Determine the y-intercept (b) of the linear function:
- The linear function is given by the equation:
[tex]\[
y = mx + b
\][/tex]
- Using one of the points, for example, (1, 3), and the calculated slope [tex]\(m = 6\)[/tex]:
[tex]\[
3 = 6 \cdot 1 + b
\][/tex]
- Solving for [tex]\(b\)[/tex]:
[tex]\[
3 = 6 + b \implies b = 3 - 6 = -3
\][/tex]
4. Construct the linear equation with the calculated slope and y-intercept:
- The linear function is:
[tex]\[
y = 6x - 3
\][/tex]
5. Find the output for the input value 6:
- Substitute [tex]\(x = 6\)[/tex] into the linear equation:
[tex]\[
y = 6 \cdot 6 - 3 = 36 - 3 = 33
\][/tex]
Therefore, the output when the input is 6 is:
[tex]\[
\boxed{33}
\][/tex]
