QUESTION IMAGE
Question
complete the table of values for the functions $f(x) = |x - 10|$ and $g(x) = \frac{1}{2}(x - 6)^2$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \
\
| $x$ | $f(x)$ | $g(x)$ | \ |
| --- | --- | --- | \ |
| 5 | \ | ||
| 6 | \ | ||
| 7 | \ | ||
| 8 | \ | ||
| 9 | \ |
\
based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?\
$x = 7$ between $x = 7$ and $x = 8$\
$x = 8$ between $x = 8$ and $x = 9$
Part 1: Completing the table for \( f(x) = |x - 10| \) and \( g(x) = \frac{1}{2}(x - 6)^2 \)
For \( f(x) = |x - 10| \):
- Step 1: When \( x = 5 \)
Substitute \( x = 5 \) into \( f(x) \):
\( f(5) = |5 - 10| = |-5| = 5 \)
- Step 2: When \( x = 6 \)
Substitute \( x = 6 \):
\( f(6) = |6 - 10| = |-4| = 4 \)
- Step 3: When \( x = 7 \)
Substitute \( x = 7 \):
\( f(7) = |7 - 10| = |-3| = 3 \)
- Step 4: When \( x = 8 \)
Substitute \( x = 8 \):
\( f(8) = |8 - 10| = |-2| = 2 \)
- Step 5: When \( x = 9 \)
Substitute \( x = 9 \):
\( f(9) = |9 - 10| = |-1| = 1 \)
For \( g(x) = \frac{1}{2}(x - 6)^2 \):
- Step 1: When \( x = 5 \)
Substitute \( x = 5 \):
\( g(5) = \frac{1}{2}(5 - 6)^2 = \frac{1}{2}(-1)^2 = \frac{1}{2}(1) = 0.5 \)
- Step 2: When \( x = 6 \)
Substitute \( x = 6 \):
\( g(6) = \frac{1}{2}(6 - 6)^2 = \frac{1}{2}(0)^2 = 0 \)
- Step 3: When \( x = 7 \)
Substitute \( x = 7 \):
\( g(7) = \frac{1}{2}(7 - 6)^2 = \frac{1}{2}(1)^2 = \frac{1}{2}(1) = 0.5 \)
- Step 4: When \( x = 8 \)
Substitute \( x = 8 \):
\( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(2)^2 = \frac{1}{2}(4) = 2 \)
- Step 5: When \( x = 9 \)
Substitute \( x = 9 \):
\( g(9) = \frac{1}{2}(9 - 6)^2 = \frac{1}{2}(3)^2 = \frac{1}{2}(9) = 4.5 \)
Completed Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 6 | 4 | 0 |
| 7 | 3 | 0.5 |
| 8 | 2 | 2 |
| 9 | 1 | 4.5 |
Part 2: Finding where \( f(x) = g(x) \)
We analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):
- At \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f(x) > g(x) \) )
- At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) (Wait, no—wait, \( g(8) = 2 \) and \( f(8) = 2 \)? Wait, no, let’s recheck:
Wait, \( f(8) = |8 - 10| = 2 \), \( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(4) = 2 \). Wait, but the table shows \( f(8) = 2 \) and \( g(8) = 2 \), so \( x = 8 \) is a solution? Wait, no, maybe I made a mistake. Wait, let’s re-express:
Wait, the original problem’s table and the options: Let’s check the values between \( x = 7 \) and \( x = 8 \), and \( x = 8 \) and \( x = 9 \).
Wait, at \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f > g \) )
At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) ( \( f = g \) )—but wait, the options include “between \( x = 7 \) and \( x = 8 \)”, “between \( x = 8 \) and \( x = 9 \)”, “\( x = 7 \)”, “\( x = 8 \)”.
Wait, but when \( x = 8 \), \( f(8) = 2 \) and \( g(8) = 2 \), so \( x = 8 \) is a solution? But let’s check again:
Wait, \( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(4) = 2 \), and \( f(8) = |8 - 10| = 2 \). So \( f(8) = g(8) \), so \( x = 8 \) is a solution. But let’s check the options:
Options:
- \( x = 7 \)
- between \( x = 7 \) and \( x = 8 \)
- \( x = 8 \)
- between \( x = 8 \) and \( x = 9 \)
Wait, at \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f > g \) )
At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) ( \( f = g \) )
Thus, the solution is \( x = 8 \).
Final Answers:
Table Completion:
- \( x = 5 \): \( f(5) = 5 \), \( g(5) = 0.5 \)
- \( x = 6 \): \( f(6) = 4 \), \( g(6) = 0 \)
- \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \)
- \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \)
- \( x = 9 \): \( f(9) = 1 \), \( g(9) = 4.5 \)
Solution to \( f(x) = g(x) \):
\( x = 8 \) (since \( f(8) = g(8) = 2 \))
Final Answer (for the equation \( f(x) = g(x) \)):
\( \boldsymbol{x = 8} \)
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Part 1: Completing the table for \( f(x) = |x - 10| \) and \( g(x) = \frac{1}{2}(x - 6)^2 \)
For \( f(x) = |x - 10| \):
- Step 1: When \( x = 5 \)
Substitute \( x = 5 \) into \( f(x) \):
\( f(5) = |5 - 10| = |-5| = 5 \)
- Step 2: When \( x = 6 \)
Substitute \( x = 6 \):
\( f(6) = |6 - 10| = |-4| = 4 \)
- Step 3: When \( x = 7 \)
Substitute \( x = 7 \):
\( f(7) = |7 - 10| = |-3| = 3 \)
- Step 4: When \( x = 8 \)
Substitute \( x = 8 \):
\( f(8) = |8 - 10| = |-2| = 2 \)
- Step 5: When \( x = 9 \)
Substitute \( x = 9 \):
\( f(9) = |9 - 10| = |-1| = 1 \)
For \( g(x) = \frac{1}{2}(x - 6)^2 \):
- Step 1: When \( x = 5 \)
Substitute \( x = 5 \):
\( g(5) = \frac{1}{2}(5 - 6)^2 = \frac{1}{2}(-1)^2 = \frac{1}{2}(1) = 0.5 \)
- Step 2: When \( x = 6 \)
Substitute \( x = 6 \):
\( g(6) = \frac{1}{2}(6 - 6)^2 = \frac{1}{2}(0)^2 = 0 \)
- Step 3: When \( x = 7 \)
Substitute \( x = 7 \):
\( g(7) = \frac{1}{2}(7 - 6)^2 = \frac{1}{2}(1)^2 = \frac{1}{2}(1) = 0.5 \)
- Step 4: When \( x = 8 \)
Substitute \( x = 8 \):
\( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(2)^2 = \frac{1}{2}(4) = 2 \)
- Step 5: When \( x = 9 \)
Substitute \( x = 9 \):
\( g(9) = \frac{1}{2}(9 - 6)^2 = \frac{1}{2}(3)^2 = \frac{1}{2}(9) = 4.5 \)
Completed Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 6 | 4 | 0 |
| 7 | 3 | 0.5 |
| 8 | 2 | 2 |
| 9 | 1 | 4.5 |
Part 2: Finding where \( f(x) = g(x) \)
We analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):
- At \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f(x) > g(x) \) )
- At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) (Wait, no—wait, \( g(8) = 2 \) and \( f(8) = 2 \)? Wait, no, let’s recheck:
Wait, \( f(8) = |8 - 10| = 2 \), \( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(4) = 2 \). Wait, but the table shows \( f(8) = 2 \) and \( g(8) = 2 \), so \( x = 8 \) is a solution? Wait, no, maybe I made a mistake. Wait, let’s re-express:
Wait, the original problem’s table and the options: Let’s check the values between \( x = 7 \) and \( x = 8 \), and \( x = 8 \) and \( x = 9 \).
Wait, at \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f > g \) )
At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) ( \( f = g \) )—but wait, the options include “between \( x = 7 \) and \( x = 8 \)”, “between \( x = 8 \) and \( x = 9 \)”, “\( x = 7 \)”, “\( x = 8 \)”.
Wait, but when \( x = 8 \), \( f(8) = 2 \) and \( g(8) = 2 \), so \( x = 8 \) is a solution? But let’s check again:
Wait, \( g(8) = \frac{1}{2}(8 - 6)^2 = \frac{1}{2}(4) = 2 \), and \( f(8) = |8 - 10| = 2 \). So \( f(8) = g(8) \), so \( x = 8 \) is a solution. But let’s check the options:
Options:
- \( x = 7 \)
- between \( x = 7 \) and \( x = 8 \)
- \( x = 8 \)
- between \( x = 8 \) and \( x = 9 \)
Wait, at \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \) ( \( f > g \) )
At \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \) ( \( f = g \) )
Thus, the solution is \( x = 8 \).
Final Answers:
Table Completion:
- \( x = 5 \): \( f(5) = 5 \), \( g(5) = 0.5 \)
- \( x = 6 \): \( f(6) = 4 \), \( g(6) = 0 \)
- \( x = 7 \): \( f(7) = 3 \), \( g(7) = 0.5 \)
- \( x = 8 \): \( f(8) = 2 \), \( g(8) = 2 \)
- \( x = 9 \): \( f(9) = 1 \), \( g(9) = 4.5 \)
Solution to \( f(x) = g(x) \):
\( x = 8 \) (since \( f(8) = g(8) = 2 \))
Final Answer (for the equation \( f(x) = g(x) \)):
\( \boldsymbol{x = 8} \)