Question

Arrange the functions in ascending order, starting with the function that eventually has the least value and ending with the function that eventually has the greatest value.
3x + 8
4x + 3
x2 + 6
3x
4x
x2

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Answer 1

To arrange the functions in ascending order based on their values as \( x \) increases, we will analyze their growth rates as \( x \) approaches infinity.

The functions are:
1. \( 3x + 8 \)
2. \( 4x + 3 \)
3. \( x^2 + 6 \)
4. \( 3x \)
5. \( 4x \)
6. \( x^2 \)

### Step 1: Identify the Dominant Term
- Linear functions: \( 3x + 8 \), \( 4x + 3 \), \( 3x \), \( 4x \)
- Quadratic functions: \( x^2 + 6 \), \( x^2 \)

As \( x \) approaches infinity, the quadratic functions grow faster than the linear functions.

### Step 2: Compare Linear Functions
Among the linear functions, the coefficients matter for their growth rates:
- \( 3x \) has a lower coefficient than \( 4x \).
- \( 3x + 8 \) and \( 4x + 3 \) also correspond to \( 3x \) and \( 4x \) but with constant terms added. However, these constants become insignificant as \( x \) becomes very large.

Thus, in the order of growth:
- \( 3x < 4x < 3x + 8 < 4x + 3 \)

### Step 3: Include Quadratic Functions
The quadratic functions \( x^2 \) and \( x^2 + 6 \) will eventually outgrow all linear functions:
- \( x^2 < x^2 + 6 \)

### Final Order
Combining all the observations, the ascending order of the functions from least to greatest value for large \( x \) is:
1. \( 3x \)
2. \( 4x \)
3. \( 3x + 8 \)
4. \( 4x + 3 \)
5. \( x^2 \)
6. \( x^2 + 6 \)

So, the final arrangement is:
1. \( 3x \)
2. \( 4x \)
3. \( 3x + 8 \)
4. \( 4x + 3 \)
5. \( x^2 \)
6. \( x^2 + 6 \)