The function \(\mathrm{t(h) = 4h - 3}\) gives the temperature in degrees Fahrenheit h hours after 6 AM. At what...

1. INFER what the problem is asking

• We know the temperature function: \(\mathrm{t(h) = 4h - 3}\)
• We want to find when the temperature equals 21°F
• This means we need to find the value of h that makes \(\mathrm{t(h) = 21}\)

2. TRANSLATE the condition into an equation

• "Temperature is 21°F" means: \(\mathrm{t(h) = 21}\)
• Substitute the function: \(\mathrm{4h - 3 = 21}\)
• Now we have a linear equation to solve

3. SIMPLIFY by solving the equation

• Add 3 to both sides: \(\mathrm{4h - 3 + 3 = 21 + 3}\)
• This gives us: \(\mathrm{4h = 24}\)
• Divide both sides by 4: \(\mathrm{h = 6}\)

Answer: 6 (hours after 6 AM)

Note: Acceptable answer forms include: 6, 6 hours, or 6 hours after 6 AM

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that finding "when the temperature is 21°F" requires setting up an equation where \(\mathrm{t(h) = 21}\). Instead, they might try to substitute 21 for h in the function, calculating \(\mathrm{t(21) = 4(21) - 3 = 81}\), which gives them a temperature value rather than a time. This fundamental misunderstanding of what the problem is asking leads to confusion and incorrect numerical answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{4h - 3 = 21}\) but make arithmetic errors during the solving process. Common mistakes include: adding 3 incorrectly (getting \(\mathrm{4h = 23}\) instead of \(\mathrm{4h = 24}\)), or dividing incorrectly (getting \(\mathrm{h = 5}\) or \(\mathrm{h = 7}\)). These calculation errors lead to wrong final answers even though the approach was correct.

The Bottom Line:

This problem requires students to work backwards from an output value to find the corresponding input value. Success depends on understanding that function problems can ask you to find either direction - given input find output, or given output find input.