Question

Complete the table of values for the functions f(x)=|x|+3 and g(x)=3x–5.
Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
x f(x) g(x)
–
1
0
1
2
3
Based on the values in the table, where does the equation f(x)=g(x) have a solution?
x=1
between x=1 and x=2
x=2
between x=2 and x=3

Answer 1

To complete the table, we need to calculate the values of \( f(x) = |x| + 3 \) and \( g(x) = 3x - 5 \) for the given \( x \) values (–1, 0, 1, 2, 3).

### Step 1: Calculate \( f(x) \) and \( g(x) \)

1. **When \( x = -1 \):**
- \( f(-1) = |-1| + 3 = 1 + 3 = 4 \)
- \( g(-1) = 3(-1) - 5 = -3 - 5 = -8 \)

2. **When \( x = 0 \):**
- \( f(0) = |0| + 3 = 0 + 3 = 3 \)
- \( g(0) = 3(0) - 5 = 0 - 5 = -5 \)

3. **When \( x = 1 \):**
- \( f(1) = |1| + 3 = 1 + 3 = 4 \)
- \( g(1) = 3(1) - 5 = 3 - 5 = -2 \)

4. **When \( x = 2 \):**
- \( f(2) = |2| + 3 = 2 + 3 = 5 \)
- \( g(2) = 3(2) - 5 = 6 - 5 = 1 \)

5. **When \( x = 3 \):**
- \( f(3) = |3| + 3 = 3 + 3 = 6 \)
- \( g(3) = 3(3) - 5 = 9 - 5 = 4 \)

### Step 2: Fill in the table

| \( x \) | \( f(x) \) | \( g(x) \) |
|---------|------------|------------|
| -1 | 4 | -8 |
| 0 | 3 | -5 |
| 1 | 4 | -2 |
| 2 | 5 | 1 |
| 3 | 6 | 4 |

### Step 3: Determine where \( f(x) = g(x) \)

To find where \( f(x) = g(x) \), we'll compare the values:

- At \( x = 1 \):
- \( f(1) = 4 \)
- \( g(1) = -2 \)

- At \( x = 2 \):
- \( f(2) = 5 \)
- \( g(2) = 1 \)

Since \( f(1) > g(1) \) and \( f(2) > g(2) \), we can see:
- At \( x = 1 \), \( f(1) > g(1) \)
- At \( x = 2 \), \( f(2) > g(2) \)

Since \( f(1) = 4\) and \( g(1) = -2 \) and \( f(2) = 5\) and \( g(2) = 1\), we can also note:
- \( g(1) \) to \( g(2) \) (too negative to positive) indicates increasing behavior.

Therefore, there must be a solution between \( x = 1 \) and \( x = 2 \) where \( f(x) = g(x) \).

### Final Answer:

The answer is **between \( x = 1 \) and \( x = 2 \)**.