To address the function [tex]\( f(x) = 2x - 6 \)[/tex] within the interval [tex]\( 3 \leq x \leq 7 \)[/tex], we will go through the points step-by-step:
1. Calculate the value of the function at the boundary points:
- Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 2 \cdot 3 - 6 = 6 - 6 = 0
\][/tex]
Thus, [tex]\( f(3) = 0 \)[/tex].
- Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = 2 \cdot 7 - 6 = 14 - 6 = 8
\][/tex]
Thus, [tex]\( f(7) = 8 \)[/tex].
2. Generate the x-values in the interval:
The interval provided is [tex]\( 3 \leq x \leq 7 \)[/tex]. The x-values within this interval (including the boundaries) are:
[tex]\[
x = 3, 4, 5, 6, 7
\][/tex]
3. Calculate the corresponding y-values:
Next, calculate [tex]\( f(x) \)[/tex] for each of these x-values:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 0
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 2 \cdot 4 - 6 = 8 - 6 = 2
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
f(5) = 2 \cdot 5 - 6 = 10 - 6 = 4
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = 2 \cdot 6 - 6 = 12 - 6 = 6
\][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = 8
\][/tex]
Therefore, the corresponding y-values are:
[tex]\[
y = 0, 2, 4, 6, 8
\][/tex]
4. Summarize the results:
We have calculated the function values at the boundary points [tex]\( x = 3 \)[/tex] and [tex]\( x = 7 \)[/tex], and the complete set of function values within the interval.
Final Results:
- [tex]\( f(3) = 0 \)[/tex]
- [tex]\( f(7) = 8 \)[/tex]
- Interval x-values: [tex]\( [3, 4, 5, 6, 7] \)[/tex]
- Corresponding y-values: [tex]\( [0, 2, 4, 6, 8] \)[/tex]
