The functions f and g are defined as \(\mathrm{f(x) = \frac{1}{4}x - 9}\) and \(\mathrm{g(x) = \frac{3}{4}x + 21}\). If...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = \frac{1}{4}x - 9}\)
  • \(\mathrm{g(x) = \frac{3}{4}x + 21}\)
  • \(\mathrm{h(x) = f(x) + g(x)}\)
• Need to find: x-coordinate of x-intercept of \(\mathrm{y = h(x)}\)

2. TRANSLATE the function addition

• Since \(\mathrm{h(x) = f(x) + g(x)}\), substitute the expressions:
  \(\mathrm{h(x) = (\frac{1}{4}x - 9) + (\frac{3}{4}x + 21)}\)

3. SIMPLIFY by combining like terms

• Combine the x terms: \(\mathrm{\frac{1}{4}x + \frac{3}{4}x = \frac{4}{4}x = x}\)
• Combine the constants: \(\mathrm{-9 + 21 = 12}\)
• Result: \(\mathrm{h(x) = x + 12}\)

4. INFER what x-intercept means

• The x-intercept occurs where the graph crosses the x-axis
• This happens when \(\mathrm{y = 0}\)
• So we need to solve: \(\mathrm{0 = x + 12}\)

5. SIMPLIFY to find the x-coordinate

\(\mathrm{0 = x + 12}\)
\(\mathrm{x = -12}\)

Answer: -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students might incorrectly add the functions by trying to multiply them instead, writing \(\mathrm{h(x) = f(x) \times g(x)}\), or get confused about what '\(\mathrm{f(x) + g(x)}\)' means in practice.

This conceptual confusion about function operations leads them to create an entirely wrong expression for \(\mathrm{h(x)}\), making it impossible to find the correct x-intercept. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{h(x) = (\frac{1}{4}x - 9) + (\frac{3}{4}x + 21)}\) but make arithmetic errors when combining like terms, such as getting \(\mathrm{\frac{1}{4} + \frac{3}{4} = \frac{4}{7}}\) instead of 1, or incorrectly combining the constants.

These calculation errors result in a wrong simplified form of \(\mathrm{h(x)}\), which then leads to an incorrect x-intercept value.

The Bottom Line:

This problem tests whether students can systematically combine functions and then apply the definition of x-intercept. The key challenge is maintaining accuracy through multiple algebraic steps while keeping track of what each step accomplishes toward the final goal.