The function f is defined by \(\mathrm{f(x) = 5x + 12}\). The function g is the inverse of f. In...

1. INFER the key property

• The problem states that "a function and its inverse always intersect on the line \(\mathrm{y = x}\) when such an intersection exists"
• This means at the intersection point, the x-coordinate equals the y-coordinate
• Instead of finding \(\mathrm{g(x)}\) explicitly, we can use this property directly

2. TRANSLATE this insight into an equation

• Since the intersection occurs where \(\mathrm{y = x}\), we need \(\mathrm{f(x) = x}\)
• Set up: \(\mathrm{5x + 12 = x}\)

3. SIMPLIFY the linear equation

• Subtract x from both sides: \(\mathrm{5x - x + 12 = 0}\)
• Combine like terms: \(\mathrm{4x + 12 = 0}\)
• Subtract 12: \(\mathrm{4x = -12}\)
• Divide by 4: \(\mathrm{x = -3}\)

4. INFER the intersection point coordinates

• Since the intersection is on \(\mathrm{y = x}\), when \(\mathrm{x = -3}\), we have \(\mathrm{y = -3}\)
• Therefore \(\mathrm{(a, b) = (-3, -3)}\)
• This means \(\mathrm{a = -3}\) and \(\mathrm{b = -3}\)

5. Calculate the final answer

• \(\mathrm{a + b = -3 + (-3) = -6}\)

Answer: A) -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students miss the hint about intersection on \(\mathrm{y = x}\) and instead try to find the inverse function \(\mathrm{g(x)}\) explicitly, then set \(\mathrm{f(x) = g(x)}\).

They would find \(\mathrm{g(x) = \frac{x-12}{5}}\), then solve:
\(\mathrm{5x + 12 = \frac{x-12}{5}}\)

This leads to a more complex equation: \(\mathrm{5(5x + 12) = x - 12}\), or \(\mathrm{25x + 60 = x - 12}\), giving \(\mathrm{24x = -72}\), so \(\mathrm{x = -3}\). While this eventually gives the same x-value, it's unnecessarily complicated and more prone to algebraic errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when solving \(\mathrm{5x + 12 = x}\), such as:

• Incorrectly combining terms: \(\mathrm{5x + 12 = x \rightarrow 6x = 12 \rightarrow x = 2}\)
• Sign errors when moving terms: \(\mathrm{5x + 12 = x \rightarrow 5x = x + 12 \rightarrow 4x = 12 \rightarrow x = 3}\)

These errors lead them to select Choice B (-3) or Choice C (6) respectively.

The Bottom Line:

This problem tests whether students can recognize and apply the geometric property that function-inverse pairs intersect on \(\mathrm{y = x}\), rather than getting bogged down in algebraic manipulation of inverse functions.