The function f is defined by \(\mathrm{f(x) = x^3 + 15}\). If \(\mathrm{f(a) = 23}\), what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^3 + 15}\)
  • Condition: \(\mathrm{f(a) = 23}\)
  • Find: the value of a

• What this tells us: We need to find the input value that makes the function output equal 23

2. TRANSLATE the condition into an equation

• Since \(\mathrm{f(a) = 23}\), we substitute a into our function:
  \(\mathrm{f(a) = a^3 + 15 = 23}\)

• This gives us the equation: \(\mathrm{a^3 + 15 = 23}\)

3. SIMPLIFY to solve for a³

• Subtract 15 from both sides:
  \(\mathrm{a^3 = 23 - 15 = 8}\)

4. SIMPLIFY to find a

• Take the cube root of both sides:
  \(\mathrm{a = \sqrt[3]{8} = 2}\)

• Check: Since \(\mathrm{2^3 = 8}\), this confirms \(\mathrm{a = 2}\)

5. Verify the answer

• \(\mathrm{f(2) = 2^3 + 15 = 8 + 15 = 23}\) ✓

Answer: C (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Not recognizing that \(\mathrm{\sqrt[3]{8} = 2}\)

Students correctly set up \(\mathrm{a^3 = 8}\) but then struggle to evaluate the cube root. They may not immediately recall that \(\mathrm{2^3 = 8}\), leading them to either guess among the answer choices or attempt incorrect calculations like thinking \(\mathrm{\sqrt[3]{8} = 1}\) or 4.

This may lead them to select Choice B (1) or cause confusion and random guessing among the given options.

The Bottom Line:

This problem tests whether students can fluently work with function notation and cube roots. The algebraic setup is straightforward, but success depends on recognizing the cube root relationship \(\mathrm{\sqrt[3]{8} = 2}\).