The function f is defined by \(\mathrm{f(x) = 5 × 2^x}\). What is the value of \(\mathrm{f(3)}\)? 10 15 40...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = 5 × 2^x}\) and we need to find \(\mathrm{f(3)}\)
• This means substitute \(\mathrm{x = 3}\) everywhere we see x in the function

2. INFER the solution approach

• We need to substitute first, then follow order of operations
• Since we have both an exponent and multiplication, evaluate the exponent first

3. Make the substitution

\(\mathrm{f(3) = 5 × 2^3}\)

4. SIMPLIFY using order of operations

• First, evaluate the exponent: \(\mathrm{2^3 = 2 × 2 × 2 = 8}\)
• Then multiply: \(\mathrm{5 × 8 = 40}\)

Answer: C (40)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak conceptual knowledge of exponential notation: Students confuse \(\mathrm{2^3}\) with \(\mathrm{2 × 3 = 6}\), or think \(\mathrm{2^3}\) just equals 2.

If they think \(\mathrm{2^3 = 2}\), they calculate: \(\mathrm{f(3) = 5 × 2 = 10}\)
This leads them to select Choice A (10)

Second Most Common Error:

Poor INFER reasoning about the exponent: Students might interpret the problem as asking them to use the number 3 directly instead of as an exponent.

They might calculate \(\mathrm{5 × 3 = 15}\), thinking the "3" in \(\mathrm{f(3)}\) should be multiplied by 5.
This may lead them to select Choice B (15)

The Bottom Line:

This problem tests whether students truly understand both function notation and exponential expressions. The key insight is recognizing that \(\mathrm{f(3)}\) requires substitution, and \(\mathrm{2^3}\) requires proper exponent evaluation, not just multiplication by 3.