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.
5x + 3
8^x + 3
x^2 + 4
5x
7^x
x^2

Explanation:

To arrange the functions in ascending order based on their eventual (as \( x \) becomes very large) values, we analyze the growth rates of different types of functions:

Step 1: Identify Function Types and Growth Rates

• Linear Functions: \( 5x \), \( 5x + 3 \) (linear, degree 1)
• Quadratic Functions: \( x^2 \), \( x^2 + 4 \) (quadratic, degree 2)
• Exponential Functions: \( 7^x \), \( 8^x + 3 \) (exponential, base > 1)

Step 2: Compare Growth Rates

• Linear vs Quadratic vs Exponential:
• Linear functions grow slower than quadratic functions.
• Quadratic functions grow slower than exponential functions.

Step 3: Compare Within Each Type

• Linear Functions:
• \( 5x \) and \( 5x + 3 \): For large \( x \), the \( +3 \) is negligible. So \( 5x < 5x + 3 \).
• Quadratic Functions:
• \( x^2 \) and \( x^2 + 4 \): For large \( x \), the \( +4 \) is negligible. So \( x^2 < x^2 + 4 \).
• Exponential Functions:
• \( 7^x \) and \( 8^x + 3 \): Since \( 8 > 7 \), \( 7^x < 8^x + 3 \) for large \( x \).

Step 4: Order All Functions

• Linear (slowest) → Quadratic → Exponential (fastest)
• Within linear: \( 5x < 5x + 3 \)
• Within quadratic: \( x^2 < x^2 + 4 \)
• Within exponential: \( 7^x < 8^x + 3 \)

Putting it all together (from least to greatest):
\( 5x \), \( 5x + 3 \), \( x^2 \), \( x^2 + 4 \), \( 7^x \), \( 8^x + 3 \)

Answer:

The functions in ascending order (from least to greatest eventual value) are:
\( 5x \), \( 5x + 3 \), \( x^2 \), \( x^2 + 4 \), \( 7^x \), \( 8^x + 3 \)