The function f is defined by \(\mathrm{f(x) = mx + b}\), where m and b are constants. If \(\mathrm{f(0) =...

1. TRANSLATE the function information

• Given information:
  • \(\mathrm{f(x) = mx + b}\) (linear function with unknown constants m and b)
  • \(\mathrm{f(0) = 18}\) (when x = 0, the function output is 18)
  • \(\mathrm{f(1) = 20}\) (when x = 1, the function output is 20)
• We need to find the value of m (the slope)

2. INFER the most efficient approach

• Since we have two function values, we can substitute them into \(\mathrm{f(x) = mx + b}\) to create two equations
• Strategy: Find b first using \(\mathrm{f(0) = 18}\), then use \(\mathrm{f(1) = 20}\) to find m

3. SIMPLIFY to find b using f(0) = 18

• Substitute x = 0 into \(\mathrm{f(x) = mx + b}\):
  \(\mathrm{f(0) = m(0) + b = 0 + b = b}\)
• Since \(\mathrm{f(0) = 18}\), we get: \(\mathrm{b = 18}\)

4. SIMPLIFY to find m using f(1) = 20

• Now substitute x = 1 into \(\mathrm{f(x) = mx + b}\):
  \(\mathrm{f(1) = m(1) + b = m + b}\)
• We know \(\mathrm{b = 18}\) and \(\mathrm{f(1) = 20}\), so: \(\mathrm{m + 18 = 20}\)
• Solving for m: \(\mathrm{m = 20 - 18 = 2}\)

5. Verify the solution

• Our function is \(\mathrm{f(x) = 2x + 18}\)
• Check: \(\mathrm{f(0) = 2(0) + 18 = 18}\) ✓
• Check: \(\mathrm{f(1) = 2(1) + 18 = 20}\) ✓

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert function notation into workable equations. They might see \(\mathrm{f(0) = 18}\) and not realize this means "substitute 0 for x and the result is 18." Instead, they might try to work with the function symbolically without using the specific values given.

This leads to confusion and guessing rather than systematic problem-solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make arithmetic errors. For example, when solving \(\mathrm{m + 18 = 20}\), they might subtract incorrectly and get \(\mathrm{m = 38}\) instead of \(\mathrm{m = 2}\).

This type of careless error can completely derail an otherwise correct approach.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and practical equation-solving. The key insight is recognizing that function values give you specific points that must satisfy the linear equation, making this a straightforward substitution problem rather than something more complex.