complete the table of values for the functions $f(x) = -2x + 6$ and $g(x) = 2(x - 1)^2$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \
$\begin{tabular}{|c|c|c|} \\hline $x$ & $f(x)$ & $g(x)$ \\\\ \\hline 0 & \\square & \\square \\\\ \\hline 1 & \\square & \\square \\\\ \\hline 2 & \\square & \\square \\\\ \\hline 3 & \\square & \\square \\\\ \\hline 4 & \\square & \\square \\\\ \\hline \\end{tabular}$
based on the values in the table, where does the equation $f(x) = g(x)$ have a solution? between $x = 0$ and $x = 1$ $x = 1$ between $x = 1$ and $x = 2$ $x = 2$

Question

complete the table of values for the functions $f(x) = -2x + 6$ and $g(x) = 2(x - 1)^2$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \

\begin{tabular}{|c|c|c|} \\hline $x$ & $f(x)$ & $g(x)$ \\\\ \\hline 0 & \\square & \\square \\\\ \\hline 1 & \\square & \\square \\\\ \\hline 2 & \\square & \\square \\\\ \\hline 3 & \\square & \\square \\\\ \\hline 4 & \\square & \\square \\\\ \\hline \\end{tabular}

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution? between $x = 0$ and $x = 1$ $x = 1$ between $x = 1$ and $x = 2$ $x = 2$

Accepted Answer

### Part 1: Completing the table for $ f(x) = -2x + 6 $ and $ g(x) = 2(x - 1)^2 $

#### For $ f(x) = -2x + 6 $:
- **When $ x = 0 $**:
  ## Step 1: Substitute $ x = 0 $ into $ f(x) $
  $ f(0) = -2(0) + 6 $
  ## Step 2: Simplify
  $ f(0) = 0 + 6 = 6 $

- **When $ x = 1 $**:
  ## Step 1: Substitute $ x = 1 $ into $ f(x) $
  $ f(1) = -2(1) + 6 $
  ## Step 2: Simplify
  $ f(1) = -2 + 6 = 4 $

- **When $ x = 2 $**:
  ## Step 1: Substitute $ x = 2 $ into $ f(x) $
  $ f(2) = -2(2) + 6 $
  ## Step 2: Simplify
  $ f(2) = -4 + 6 = 2 $

- **When $ x = 3 $**:
  ## Step 1: Substitute $ x = 3 $ into $ f(x) $
  $ f(3) = -2(3) + 6 $
  ## Step 2: Simplify
  $ f(3) = -6 + 6 = 0 $

- **When $ x = 4 $**:
  ## Step 1: Substitute $ x = 4 $ into $ f(x) $
  $ f(4) = -2(4) + 6 $
  ## Step 2: Simplify
  $ f(4) = -8 + 6 = -2 $

#### For $ g(x) = 2(x - 1)^2 $:
- **When $ x = 0 $**:
  ## Step 1: Substitute $ x = 0 $ into $ g(x) $
  $ g(0) = 2(0 - 1)^2 $
  ## Step 2: Simplify the exponent
  $ g(0) = 2(-1)^2 = 2(1) = 2 $

- **When $ x = 1 $**:
  ## Step 1: Substitute $ x = 1 $ into $ g(x) $
  $ g(1) = 2(1 - 1)^2 $
  ## Step 2: Simplify the exponent
  $ g(1) = 2(0)^2 = 2(0) = 0 $

- **When $ x = 2 $**:
  ## Step 1: Substitute $ x = 2 $ into $ g(x) $
  $ g(2) = 2(2 - 1)^2 $
  ## Step 2: Simplify the exponent
  $ g(2) = 2(1)^2 = 2(1) = 2 $

- **When $ x = 3 $**:
  ## Step 1: Substitute $ x = 3 $ into $ g(x) $
  $ g(3) = 2(3 - 1)^2 $
  ## Step 2: Simplify the exponent
  $ g(3) = 2(2)^2 = 2(4) = 8 $

- **When $ x = 4 $**:
  ## Step 1: Substitute $ x = 4 $ into $ g(x) $
  $ g(4) = 2(4 - 1)^2 $
  ## Step 2: Simplify the exponent
  $ g(4) = 2(3)^2 = 2(9) = 18 $

### Filled Table:
| $ x $ | $ f(x) $ | $ g(x) $ |
|--------|------------|------------|
| 0      | 6          | 2          |
| 1      | 4          | 0          |
| 2      | 2          | 2          |
| 3      | 0          | 8          |
| 4      | -2         | 18         |

### Part 2: Finding where $ f(x) = g(x) $
We look for intervals where $ f(x) $ and $ g(x) $ cross (i.e., where $ f(x) $ and $ g(x) $ change from $ f(x) > g(x) $ to $ f(x) < g(x) $ or vice versa).

- At $ x = 0 $: $ f(0) = 6 $, $ g(0) = 2 $ (so $ f(x) > g(x) $)
- At $ x = 1 $: $ f(1) = 4 $, $ g(1) = 0 $ (so $ f(x) > g(x) $)
- At $ x = 2 $: $ f(2) = 2 $, $ g(2) = 2 $ (Wait, actually, let's check the values again. Wait, at $ x=1 $, $ f(1)=4 $, $ g(1)=0 $; at $ x=2 $, $ f(2)=2 $, $ g(2)=2 $? Wait, no, wait: $ g(2) = 2(2-1)^2 = 2(1) = 2 $, and $ f(2) = -2(2) +6 = 2 $. Wait, but let's check the interval between $ x=1 $ and $ x=2 $:

Wait, maybe I made a mistake. Wait, let's re-express the functions:

$ f(x) = -2x +6 $ (linear, decreasing)

$ g(x) = 2(x-1)^2 $ (parabola opening upwards, vertex at $ (1, 0) $)

At $ x=1 $: $ f(1)=4 $, $ g(1)=0 $ (so $ f > g $)

At $ x=2 $: $ f(2)=2 $, $ g(2)=2 $ (so $ f = g $ at $ x=2 $? Wait, but the options include "between $ x=1 $ and $ x=2 $" or $ x=2 $. Wait, let's check the table again:

Wait, when $ x=1 $: $ f(1)=4 $, $ g(1)=0 $

When $ x=2 $: $ f(2)=2 $, $ g(2)=2 $

Wait, but the problem's options: "between $ x=0 $ and $ x=1 $", "between $ x=1 $ and $ x=2 $", "x=1", "x=2".

Wait, but let's check the values:

At $ x=1 $: $ f(1)=4 $, $ g(1)=0 $ (f > g)

At $ x=2 $: $ f(2)=2 $, $ g(2)=2 $ (f = g)

Wait, but maybe I miscalculated $ g(2) $. Wait, $ g(2) = 2(2-1)^2 = 2(1) = 2 $, and $ f(2) = -2(2) +6 = 2 $. So $ f(2) = g(2) $. But let's check the interval between $ x=1 $ and $ x=2 $:

Wait, maybe the original problem's table has a typo, but according to the calculations:

Wait, no, let's re-express:

Wait, $ f(x) = -2x +6 $

$ g(x) = 2(x-1)^2 $

Let's solve $ -2x +6 = 2(x-1)^2 $

$ -2x +6 = 2(x^2 - 2x +1) $

$ -2x +6 = 2x^2 -4x +2 $

$ 0 = 2x^2 -2x -4 $

$ 0 = x^2 -x -2 $

$ 0 = (x - 2)(x +1) $

So solutions at $ x=2 $ and $ x=-1 $. So in the interval $ x=1 $ to $ x=2 $, the functions cross? Wait, no, at $ x=2 $, they are equal. But according to the table:

At $ x=1 $: $ f=4 $, $ g=0 $

At $ x=2 $: $ f=2 $, $ g=2 $

So between $ x=1 $ and $ x=2 $, $ f(x) $ decreases from 4 to 2, and $ g(x) $ increases from 0 to 2. So they meet at $ x=2 $. But let's check the options. The options are:

- between $ x=0 $ and $ x=1 $
- between $ x=1 $ and $ x=2 $
- $ x=1 $
- $ x=2 $

Wait, according to the table, at $ x=2 $, $ f(x)=g(x)=2 $, so the solution is $ x=2 $? But let's check the calculations again.

Wait, $ f(2) = -2(2) +6 = 2 $

$ g(2) = 2(2-1)^2 = 2(1) = 2 $

So $ f(2) = g(2) $, so the solution is $ x=2 $. But let's check the interval between $ x=1 $ and $ x=2 $:

At $ x=1 $, $ f(1)=4 $, $ g(1)=0 $ (f > g)

At $ x=2 $, $ f(2)=2 $, $ g(2)=2 $ (f = g)

So the solution is at $ x=2 $, or between $ x=1 $ and $ x=2 $? Wait, the equation $ f(x) = g(x) $ has a solution at $ x=2 $, but let's check the options. The options include "x=2" as a choice.

But let's re-examine the table:

Wait, maybe I made a mistake in $ g(2) $. Wait, $ g(2) = 2(2-1)^2 = 2(1) = 2 $, correct. $ f(2) = -2(2) +6 = 2 $, correct. So $ f(2) = g(2) $, so the solution is $ x=2 $. But let's check the options. The options are:

- between $ x=0 $ and $ x=1 $
- between $ x=1 $ and $ x=2 $
- $ x=1 $
- $ x=2 $

So the correct answer for the solution of $ f(x) = g(x) $ is $ x=2 $? Wait, but let's check the original problem's options. Wait, the problem says "Based on the values in the table, where does the equation $ f(x) = g(x) $ have a solution?"

Looking at the table:

At $ x=1 $: $ f=4 $, $ g=0 $ (f > g)

At $ x=2 $: $ f=2 $, $ g=2 $ (f = g)

So the solution is at $ x=2 $. But let's confirm with the equation:

$ -2x +6 = 2(x-1)^2 $

$ -2x +6 = 2(x^2 - 2x +1) $

$ -2x +6 = 2x^2 -4x +2 $

$ 0 = 2x^2 -2x -4 $

$ 0 = x^2 -x -2 $

$ 0 = (x - 2)(x +1) $

So solutions at $ x=2 $ and $ x=-1 $. So in the table's range (x=0 to 4), the solution is at $ x=2 $.

### Final Table:
| $ x $ | $ f(x) $ | $ g(x) $ |
|--------|------------|------------|
| 0      | 6          | 2          |
| 1      | 4          | 0          |
| 2      | 2          | 2          |
| 3      | 0          | 8          |
| 4      | -2         | 18         |

### Solution to $ f(x) = g(x) $:
From the table, at $ x=2 $, $ f(x) = g(x) = 2 $. So the solution is $ x=2 $.

### Answers:

#### Table Completion:
- $ x=0 $: $ f(0)=6 $, $ g(0)=2 $
- $ x=1 $: $ f(1)=4 $, $ g(1)=0 $
- $ x=2 $: $ f(2)=2 $, $ g(2)=2 $
- $ x=3 $: $ f(3)=0 $, $ g(3)=8 $
- $ x=4 $: $ f(4)=-2 $, $ g(4)=18 $

#### Solution to $ f(x) = g(x) $:
$ x = 2 $ (or between $ x=1 $ and $ x=2 $, but since at $ x=2 $ they are equal, the answer is $ x=2 $)

### Final Answers for the Table (filled):
| $ x $ | $ f(x) $ | $ g(x) $ |
|--------|------------|------------|
| 0      | 6          | 2          |
| 1      | 4          | 0          |
| 2      | 2          | 2          |
| 3      | 0          | 8          |
| 4      | -2         | 18         |

### Solution to $ f(x) = g(x) $:
$ \boldsymbol{x = 2} $ (or the option "x = 2" from the choices)

\( x \)	\( f(x) \)	\( g(x) \)
1	4	0
2	2	2
3	0	8
4	-2	18

\( x \)	\( f(x) \)	\( g(x) \)
1	4	0
2	2	2
3	0	8
4	-2	18

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part 1: Completing the table for \( f(x) = -2x + 6 \) and \( g(x) = 2(x - 1)^2 \)

For \( f(x) = -2x + 6 \):

Step 1: Substitute \( x = 0 \) into \( f(x) \)

Step 2: Simplify

Step 1: Substitute \( x = 1 \) into \( f(x) \)

Step 2: Simplify

Step 1: Substitute \( x = 2 \) into \( f(x) \)

Step 2: Simplify

Step 1: Substitute \( x = 3 \) into \( f(x) \)

Step 2: Simplify

Step 1: Substitute \( x = 4 \) into \( f(x) \)

Step 2: Simplify

For \( g(x) = 2(x - 1)^2 \):

Step 1: Substitute \( x = 0 \) into \( g(x) \)

Step 2: Simplify the exponent

Step 1: Substitute \( x = 1 \) into \( g(x) \)

Step 2: Simplify the exponent

Step 1: Substitute \( x = 2 \) into \( g(x) \)

Step 2: Simplify the exponent

Step 1: Substitute \( x = 3 \) into \( g(x) \)

Step 2: Simplify the exponent

Step 1: Substitute \( x = 4 \) into \( g(x) \)

Step 2: Simplify the exponent

Filled Table:

Part 2: Finding where \( f(x) = g(x) \)

Answer:

Table Completion:

Solution to \( f(x) = g(x) \):

Final Answers for the Table (filled):

Solution to \( f(x) = g(x) \):