Grade: 9 Subject: Math (Algebra I) Unit: Functions & Modeling Lesson: 5 of 6 SAT: AdvancedMath ACT: Math

Common Mistakes

Learning Objectives

In this lesson, you will:

Identify the error and find the correct answer. Click each question to reveal the solution.

Question 1: A student says f(3) = 3f for f(x) = 2x + 1. What is the error and correct answer?

Error: Treating f(3) as multiplication instead of function notation.

Correct: f(3) = 2(3) + 1 = 7. f(3) means "evaluate f at x = 3."

Question 2: For f(x) = x², a student writes f(a + b) = a² + b². What is wrong?

Error: Incorrectly distributing the squaring operation.

Correct: f(a + b) = (a + b)² = a² + 2ab + b². You must square the entire expression.

Question 3: A student claims f(x) = x + 5 and g(x) = x + 5 are different functions because they use different letters. Are they?

Answer: No, they are the same function.

Explanation: The variable is just a placeholder. Both rules say "add 5 to the input." f(2) = 7 and g(2) = 7.

Question 4: For f(x) = 3x - 2, a student says f(-4) = 3(-4) - 2 = -12 - 2 = -10. Is this correct?

Answer: Incorrect. The answer should be -14.

Correct: f(-4) = 3(-4) - 2 = -12 - 2 = -14. The student wrote -10 but should have gotten -14.

Question 5: A student writes f(2x) = 2f(x) for f(x) = x + 3. Find f(2x) correctly.

Error: Assuming function of 2x equals 2 times the function.

Correct: f(2x) = 2x + 3. Check: f(x) = x + 3, so 2f(x) = 2x + 6, which differs from 2x + 3.

Question 6: For g(x) = x² - 4, a student says g(0) = 0. What is the correct answer?

Error: Forgetting to evaluate the constant term.

Correct: g(0) = (0)² - 4 = 0 - 4 = -4.

Question 7: If f(x) = 5 - x, a student says f(-3) = 5 - (-3) = 5 - 3 = 2. What is the error?

Error: Incorrectly handling the double negative.

Correct: f(-3) = 5 - (-3) = 5 + 3 = 8. Subtracting a negative is the same as adding.

Question 8: A student confuses f(g(x)) and f(x)g(x). If f(x) = x + 1 and g(x) = 2x, find both correctly.

f(g(x)): f(2x) = 2x + 1 (composition - substitute g(x) into f)

f(x)g(x): (x + 1)(2x) = 2x² + 2x (multiplication of outputs)

Question 9: For h(x) = |x - 2|, a student says h(5) = |5| - |2| = 3. What is the error?

Error: Incorrectly splitting the absolute value.

Correct: h(5) = |5 - 2| = |3| = 3. (Same answer, but wrong method would fail for h(-1) = |-1 - 2| = |-3| = 3, not -1 - 2 = -3.)

Question 10: A student says if f(a) = f(b), then a = b for any function. Give a counterexample.

Counterexample: For f(x) = x², f(2) = 4 and f(-2) = 4, so f(2) = f(-2) but 2 ≠ -2.

Lesson: Only one-to-one functions guarantee that equal outputs mean equal inputs.