Common Mistakes
Learn to recognize and avoid the most frequent errors students make with functions and transformations.
📖 Learn
Understanding common mistakes helps you avoid them. This lesson identifies errors that appear frequently on tests and in homework, explains why they happen, and shows how to correct them.
Why Study Mistakes?
Research shows that analyzing errors is one of the most effective ways to deepen understanding. When you understand why something is wrong, you're less likely to make that error yourself.
💡 Common Errors and Corrections
Mistake 1: Confusing Horizontal Shift Direction
Wrong: "f(x + 3) shifts the graph right 3 units"
Correct: f(x + 3) shifts the graph left 3 units
Why it happens: Students assume "+ means right" but horizontal transformations work opposite to what you might expect because the change is inside the function.
Remember: f(x - h) shifts right h units; f(x + h) shifts left h units
Mistake 2: Applying Transformations in Wrong Order
Wrong: For f(x) = 2(x - 1)^2 + 3, shifting first then stretching
Correct: Apply transformations in this order:
- Horizontal shift (inside parentheses)
- Stretch/compression and reflection
- Vertical shift (outside)
Why it matters: Order affects the final position of the graph.
Mistake 3: Forgetting Domain Restrictions After Transformations
Wrong: Stating domain of g(x) = sqrt(x - 4) is all real numbers
Correct: Domain is x >= 4
Why it happens: Students apply transformations to the graph but forget to transform the domain.
Rule: If f(x) = sqrt(x) has domain x >= 0, then f(x - 4) has domain x - 4 >= 0, so x >= 4.
Mistake 4: Errors with Function Composition
Wrong: f(g(x)) = f(x) * g(x)
Correct: f(g(x)) means substitute g(x) into f
Example: If f(x) = x^2 and g(x) = x + 1
- f(g(x)) = (x + 1)^2 = x^2 + 2x + 1
- g(f(x)) = x^2 + 1
Note: f(g(x)) is NOT equal to g(f(x)) in general!
Mistake 5: Reflection vs. Negative Input
Wrong: Treating f(-x) and -f(x) as the same thing
Correct:
- f(-x) reflects over the y-axis (horizontal reflection)
- -f(x) reflects over the x-axis (vertical reflection)
Test yourself: If f(x) = x^2, then f(-x) = (-x)^2 = x^2, but -f(x) = -x^2
Mistake 6: Inverse Function Errors
Wrong: The inverse of f(x) = 2x + 3 is f^(-1)(x) = 1/(2x + 3)
Correct: f^(-1)(x) = (x - 3)/2
Why it happens: Students confuse inverse function with reciprocal. The notation f^(-1) does NOT mean 1/f.
To find inverse: Swap x and y, then solve for y.
Mistake 7: Vertical Stretch Confusion
Wrong: Thinking 2f(x) and f(2x) produce the same result
Correct:
- 2f(x) is a vertical stretch by factor 2
- f(2x) is a horizontal compression by factor 1/2
Key insight: Multiplying outside affects y-values; multiplying inside affects x-values (inversely).
Mistake 8: Even/Odd Function Errors
Wrong: Assuming all polynomials with even powers are even functions
Correct: A function is even if f(-x) = f(x) for ALL x in the domain
Example: f(x) = x^4 + x^2 is even, but g(x) = x^4 + x is neither (has odd-power term)
Test: Always substitute -x and simplify completely before deciding.
✏️ Practice: Find the Errors
Each problem contains an error. Find it and correct it.
Problem 1
Student work: "The function g(x) = (x - 2)^2 is the parent function f(x) = x^2 shifted left 2 units."
Find and correct the error.
Show Answer
Error: Direction of shift. (x - 2) shifts right 2 units, not left.
Problem 2
Student work: "If f(x) = 3x - 1 and g(x) = x^2, then f(g(2)) = f(2) * g(2) = 5 * 4 = 20"
Find and correct the error.
Show Answer
Error: Wrong definition of composition. f(g(2)) = f(4) = 3(4) - 1 = 11
Problem 3
Student work: "The domain of h(x) = sqrt(5 - x) is x >= 5"
Find and correct the error.
Show Answer
Error: Wrong inequality direction. Need 5 - x >= 0, so x <= 5
Problem 4
Student work: "To find the inverse of f(x) = x^3 + 2, I get f^(-1)(x) = 1/(x^3 + 2)"
Find and correct the error.
Show Answer
Error: Confused inverse with reciprocal. Correct: f^(-1)(x) = cuberoot(x - 2)
Problem 5
Student work: "The graph of y = 3|x| is the graph of y = |x| stretched horizontally by 3"
Find and correct the error.
Show Answer
Error: It's a vertical stretch by 3, not horizontal. Multiplying outside affects y-values.
Problem 6
Student work: "f(x) = x^3 is an even function because 3 is odd and odd functions... wait, I mean it's odd."
Verify this claim correctly.
Show Answer
f(-x) = (-x)^3 = -x^3 = -f(x), so yes, f(x) = x^3 is odd. The reasoning should use the definition, not just look at the exponent.
Problem 7
Student work: "g(x) = -f(x - 3) + 2: First shift up 2, then shift right 3, then reflect over x-axis"
Find and correct the error.
Show Answer
Error: Wrong order. Correct order: horizontal shift right 3, reflect over x-axis, then shift up 2.
Problem 8
Student work: "If f(2) = 5, then for g(x) = f(x + 1), g(2) = 5 + 1 = 6"
Find and correct the error.
Show Answer
Error: Misunderstood transformation. g(2) = f(2 + 1) = f(3), which we don't know. g(1) = f(2) = 5.
Problem 9
Student work: "The range of f(x) = x^2 - 4 is all real numbers because it's a parabola"
Find and correct the error.
Show Answer
Error: Parabolas have restricted range. This opens upward with vertex at (0, -4), so range is y >= -4.
Problem 10
Student work: "f(x) = 2^x has domain x > 0 because you can't raise 2 to a negative power"
Find and correct the error.
Show Answer
Error: You CAN raise 2 to negative powers: 2^(-1) = 1/2. Domain of f(x) = 2^x is all real numbers.
Problem 11
Student work: "The function f(x) = |x| + x is even because |x| is even"
Find and correct the error.
Show Answer
Error: Must check the whole function. f(-x) = |-x| + (-x) = |x| - x, which is not equal to f(x) = |x| + x. The function is neither even nor odd.
Problem 12
Student work: "For f(x) = sqrt(x), the graph of f(x - 2) + 3 passes through (2, 3)"
Verify or correct this claim.
Show Answer
Correct! At x = 2: sqrt(2-2) + 3 = sqrt(0) + 3 = 0 + 3 = 3. The point (2, 3) is on the graph.
✅ Check Your Understanding
Before moving on, make sure you can:
- Correctly identify horizontal shift direction from function notation
- Apply transformations in the proper order
- Find domain and range after transformations
- Compute function compositions correctly
- Distinguish between inverse functions and reciprocals
- Identify even and odd functions using definitions
- Differentiate between vertical and horizontal transformations
🚀 Next Steps
- Review any error types you found surprising
- Create a personal "error checklist" to use on tests
- Move on to the Unit Quiz to test your mastery
- Return to earlier lessons if any concepts need reinforcement