Answer :
To solve the problem of finding [tex]\( f(12) - f(2) \)[/tex] for the linear function [tex]\( f \)[/tex], given that [tex]\( f(6) - f(2) = 12 \)[/tex], we can use the properties of linear functions.
1. Linear Function Property:
A linear function is of the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. The slope [tex]\( m \)[/tex] represents the change in [tex]\( f(x) \)[/tex] per unit change in [tex]\( x \)[/tex].
2. Given Condition:
We are given [tex]\( f(6) - f(2) = 12 \)[/tex]. This means the change in the function value over the change in the input is 12 for an input change from 2 to 6.
3. Calculate the Change in Input:
- The change in input from 2 to 6 is [tex]\( 6 - 2 = 4 \)[/tex].
4. Determine the Slope:
- Since the difference [tex]\( f(6) - f(2) = 12 \)[/tex] corresponds to an input difference of 4, the slope [tex]\( m \)[/tex] can be calculated as:
[tex]\[
m = \frac{f(6) - f(2)}{6 - 2} = \frac{12}{4} = 3
\][/tex]
5. Determine [tex]\( f(12) - f(2) \)[/tex] using the Slope:
- We need to find the difference between [tex]\( f(12) \)[/tex] and [tex]\( f(2) \)[/tex]. The change in input from 2 to 12 is calculated as:
[tex]\[
12 - 2 = 10
\][/tex]
- Since the slope [tex]\( m \)[/tex] is 3, the change in [tex]\( f(x) \)[/tex] over this new input difference is:
[tex]\[
f(12) - f(2) = m \times \text{change in input} = 3 \times 10 = 30
\][/tex]
Therefore, the value of [tex]\( f(12) - f(2) \)[/tex] is [tex]\(\boxed{30}\)[/tex].
1. Linear Function Property:
A linear function is of the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. The slope [tex]\( m \)[/tex] represents the change in [tex]\( f(x) \)[/tex] per unit change in [tex]\( x \)[/tex].
2. Given Condition:
We are given [tex]\( f(6) - f(2) = 12 \)[/tex]. This means the change in the function value over the change in the input is 12 for an input change from 2 to 6.
3. Calculate the Change in Input:
- The change in input from 2 to 6 is [tex]\( 6 - 2 = 4 \)[/tex].
4. Determine the Slope:
- Since the difference [tex]\( f(6) - f(2) = 12 \)[/tex] corresponds to an input difference of 4, the slope [tex]\( m \)[/tex] can be calculated as:
[tex]\[
m = \frac{f(6) - f(2)}{6 - 2} = \frac{12}{4} = 3
\][/tex]
5. Determine [tex]\( f(12) - f(2) \)[/tex] using the Slope:
- We need to find the difference between [tex]\( f(12) \)[/tex] and [tex]\( f(2) \)[/tex]. The change in input from 2 to 12 is calculated as:
[tex]\[
12 - 2 = 10
\][/tex]
- Since the slope [tex]\( m \)[/tex] is 3, the change in [tex]\( f(x) \)[/tex] over this new input difference is:
[tex]\[
f(12) - f(2) = m \times \text{change in input} = 3 \times 10 = 30
\][/tex]
Therefore, the value of [tex]\( f(12) - f(2) \)[/tex] is [tex]\(\boxed{30}\)[/tex].
