Common Mistakes
Overview
Learn to identify and avoid common errors when working with functions. Understanding these mistakes will help you catch errors in your own work.
Practice Problems
Question 1: A student says f(x) = x² is a linear function because "x has a power of 1." What's wrong with this reasoning?
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Answer: The exponent on x is 2, not 1.
In x², the exponent is 2. Linear functions have x to the first power only (like 3x or x + 5). This is a quadratic function.
Question 2: A student found f(3) = 15 for f(x) = 2x + 3. Is this correct?
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Answer: No, f(3) = 9
f(3) = 2(3) + 3 = 6 + 3 = 9, not 15. The student likely multiplied 2 × 3 × 3 = 18 or made another error.
Question 3: A student claims {(1,3), (2,3), (3,3)} is NOT a function because "all the y-values are the same." Is this correct?
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Answer: No, this IS a function.
A function requires each x to have exactly one y. Multiple x-values can share the same y-value. This is a constant function where f(x) = 3.
Question 4: For f(x) = 5 - 2x, a student calculated f(-2) = 5 - 2(-2) = 5 - (-4) = 5 + 4 = 9. What was done correctly and what may confuse others?
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Answer: The work is actually correct!
f(-2) = 5 - 2(-2) = 5 - (-4) = 5 + 4 = 9. Subtracting a negative becomes addition. This is correct but often confuses students.
Question 5: A student wrote the slope of f(x) = 4 + 3x as 4. What's the error?
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Answer: The slope is 3, not 4.
Rewrite as f(x) = 3x + 4 (standard form mx + b). The slope m = 3, and b = 4 is the y-intercept.
Question 6: A student says f(2x) means f times 2 times x. Is this interpretation correct?
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Answer: No, this is incorrect.
f(2x) means "evaluate f at the input 2x." If f(x) = x + 3, then f(2x) = (2x) + 3 = 2x + 3. It's not multiplication of f.
Question 7: Given g(x) = x², a student calculated g(3) + g(4) = g(7) = 49. Find the error.
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Answer: g(3) + g(4) does NOT equal g(7)
g(3) + g(4) = 9 + 16 = 25. But g(7) = 49. In general, f(a) + f(b) ≠ f(a+b) for most functions.
Question 8: A student claims that if f(x) = 2x + 1 and f(a) = 7, then a = 7. Is this correct?
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Answer: No, a = 3
If 2a + 1 = 7, then 2a = 6, so a = 3. The student confused f(a) = 7 with a = 7.
Question 9: A student graphed f(x) = -x + 2 with a positive slope going up to the right. What's wrong?
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Answer: The slope should be negative (going down to the right)
The slope is -1 (negative), so the line should go down from left to right, not up.
Question 10: For f(x) = 3x - 6, a student says the x-intercept is -6. Find the actual x-intercept.
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Answer: The x-intercept is 2, not -6
Set f(x) = 0: 3x - 6 = 0, 3x = 6, x = 2. The student confused the y-intercept (-6 when in y = mx + b form) with the x-intercept.