QUESTION IMAGE
Question
complete the table of values for the functions ( f(x) = \frac{5}{2}x - 3 ) and ( g(x) = -(x - 2)^2 + 4 ).
write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
| ( x ) | ( f(x) ) | ( g(x) ) |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 |
based on the values in the table, where does the equation ( f(x) = g(x) ) have a solution?
( x = 2 ) ( quad ) between ( x = 2 ) and ( x = 3 )
( x = 3 ) ( quad ) between ( x = 3 ) and ( x = 4 )
Part 1: Completing the table for \( f(x) = \frac{5}{2}x - 3 \) and \( g(x) = -(x - 2)^2 + 4 \)
For \( f(x) = \frac{5}{2}x - 3 \):
- When \( x = 0 \):
Step 1: Substitute \( x = 0 \) into \( f(x) \)
\( f(0) = \frac{5}{2}(0) - 3 = 0 - 3 = -3 \)
- When \( x = 1 \):
Step 1: Substitute \( x = 1 \) into \( f(x) \)
\( f(1) = \frac{5}{2}(1) - 3 = \frac{5}{2} - 3 = \frac{5}{2} - \frac{6}{2} = -\frac{1}{2} = -0.5 \)
- When \( x = 2 \):
Step 1: Substitute \( x = 2 \) into \( f(x) \)
\( f(2) = \frac{5}{2}(2) - 3 = 5 - 3 = 2 \)
- When \( x = 3 \):
Step 1: Substitute \( x = 3 \) into \( f(x) \)
\( f(3) = \frac{5}{2}(3) - 3 = \frac{15}{2} - 3 = \frac{15}{2} - \frac{6}{2} = \frac{9}{2} = 4.5 \)
- When \( x = 4 \):
Step 1: Substitute \( x = 4 \) into \( f(x) \)
\( f(4) = \frac{5}{2}(4) - 3 = 10 - 3 = 7 \)
For \( g(x) = -(x - 2)^2 + 4 \):
- When \( x = 0 \):
Step 1: Substitute \( x = 0 \) into \( g(x) \)
\( g(0) = -(0 - 2)^2 + 4 = -( -2)^2 + 4 = -4 + 4 = 0 \)
- When \( x = 1 \):
Step 1: Substitute \( x = 1 \) into \( g(x) \)
\( g(1) = -(1 - 2)^2 + 4 = -( -1)^2 + 4 = -1 + 4 = 3 \)
- When \( x = 2 \):
Step 1: Substitute \( x = 2 \) into \( g(x) \)
\( g(2) = -(2 - 2)^2 + 4 = -0 + 4 = 4 \)
- When \( x = 3 \):
Step 1: Substitute \( x = 3 \) into \( g(x) \)
\( g(3) = -(3 - 2)^2 + 4 = -(1)^2 + 4 = -1 + 4 = 3 \)
- When \( x = 4 \):
Step 1: Substitute \( x = 4 \) into \( g(x) \)
\( g(4) = -(4 - 2)^2 + 4 = -(2)^2 + 4 = -4 + 4 = 0 \)
Completed Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 1 | -0.5 | 3 |
| 2 | 2 | 4 |
| 3 | 4.5 | 3 |
| 4 | 7 | 0 |
Part 2: Finding where \( f(x) = g(x) \)
We analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):
- At \( x = 2 \): \( f(2) = 2 \), \( g(2) = 4 \) ( \( f(x) < g(x) \) )
- At \( x = 3 \): \( f(3) = 4.5 \), \( g(3) = 3 \) ( \( f(x) > g(x) \) )
Since \( f(x) \) goes from less than \( g(x) \) at \( x = 2 \) to greater than \( g(x) \) at \( x = 3 \), by the Intermediate Value Theorem, there is a solution to \( f(x) = g(x) \) between \( x = 2 \) and \( x = 3 \).
Final Answers:
Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 1 | \(-\frac{1}{2}\) (or \(-0.5\)) | \(3\) |
| 2 | \(2\) | \(4\) |
| 3 | \(\frac{9}{2}\) (or \(4.5\)) | \(3\) |
| 4 | \(7\) | \(0\) |
Solution to \( f(x) = g(x) \):
The equation \( f(x) = g(x) \) has a solution between \( x = 2 \) and \( x = 3 \).
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Part 1: Completing the table for \( f(x) = \frac{5}{2}x - 3 \) and \( g(x) = -(x - 2)^2 + 4 \)
For \( f(x) = \frac{5}{2}x - 3 \):
- When \( x = 0 \):
Step 1: Substitute \( x = 0 \) into \( f(x) \)
\( f(0) = \frac{5}{2}(0) - 3 = 0 - 3 = -3 \)
- When \( x = 1 \):
Step 1: Substitute \( x = 1 \) into \( f(x) \)
\( f(1) = \frac{5}{2}(1) - 3 = \frac{5}{2} - 3 = \frac{5}{2} - \frac{6}{2} = -\frac{1}{2} = -0.5 \)
- When \( x = 2 \):
Step 1: Substitute \( x = 2 \) into \( f(x) \)
\( f(2) = \frac{5}{2}(2) - 3 = 5 - 3 = 2 \)
- When \( x = 3 \):
Step 1: Substitute \( x = 3 \) into \( f(x) \)
\( f(3) = \frac{5}{2}(3) - 3 = \frac{15}{2} - 3 = \frac{15}{2} - \frac{6}{2} = \frac{9}{2} = 4.5 \)
- When \( x = 4 \):
Step 1: Substitute \( x = 4 \) into \( f(x) \)
\( f(4) = \frac{5}{2}(4) - 3 = 10 - 3 = 7 \)
For \( g(x) = -(x - 2)^2 + 4 \):
- When \( x = 0 \):
Step 1: Substitute \( x = 0 \) into \( g(x) \)
\( g(0) = -(0 - 2)^2 + 4 = -( -2)^2 + 4 = -4 + 4 = 0 \)
- When \( x = 1 \):
Step 1: Substitute \( x = 1 \) into \( g(x) \)
\( g(1) = -(1 - 2)^2 + 4 = -( -1)^2 + 4 = -1 + 4 = 3 \)
- When \( x = 2 \):
Step 1: Substitute \( x = 2 \) into \( g(x) \)
\( g(2) = -(2 - 2)^2 + 4 = -0 + 4 = 4 \)
- When \( x = 3 \):
Step 1: Substitute \( x = 3 \) into \( g(x) \)
\( g(3) = -(3 - 2)^2 + 4 = -(1)^2 + 4 = -1 + 4 = 3 \)
- When \( x = 4 \):
Step 1: Substitute \( x = 4 \) into \( g(x) \)
\( g(4) = -(4 - 2)^2 + 4 = -(2)^2 + 4 = -4 + 4 = 0 \)
Completed Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 1 | -0.5 | 3 |
| 2 | 2 | 4 |
| 3 | 4.5 | 3 |
| 4 | 7 | 0 |
Part 2: Finding where \( f(x) = g(x) \)
We analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):
- At \( x = 2 \): \( f(2) = 2 \), \( g(2) = 4 \) ( \( f(x) < g(x) \) )
- At \( x = 3 \): \( f(3) = 4.5 \), \( g(3) = 3 \) ( \( f(x) > g(x) \) )
Since \( f(x) \) goes from less than \( g(x) \) at \( x = 2 \) to greater than \( g(x) \) at \( x = 3 \), by the Intermediate Value Theorem, there is a solution to \( f(x) = g(x) \) between \( x = 2 \) and \( x = 3 \).
Final Answers:
Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 1 | \(-\frac{1}{2}\) (or \(-0.5\)) | \(3\) |
| 2 | \(2\) | \(4\) |
| 3 | \(\frac{9}{2}\) (or \(4.5\)) | \(3\) |
| 4 | \(7\) | \(0\) |
Solution to \( f(x) = g(x) \):
The equation \( f(x) = g(x) \) has a solution between \( x = 2 \) and \( x = 3 \).