Decision and conditions
Track composed domains, operation grouping, cumulative price tiers and reverse sign-constrained processes. A defined operation need not be associative; cumulative tiers charge each portion at its own rate.
Worked example
Let f(x)=1/(x−3). What is the sum of all real inputs x for which f(f(x)) is undefined? The inner function is undefined at x=3. The outer function also fails when 1/(x−3)=3, giving x=3+1/3. The two excluded inputs sum to 19/3. Method choice: Check the inner denominator first, then check when the inner output makes the outer denominator zero.
Contrast: change the reasoning
For real a and b define a ⊙ b = a+2b−2ab. What is (3 ⊙ 2) ⊙ 3 − 3 ⊙ (2 ⊙ 3)? 3 ⊙ 2=-5, so (3 ⊙ 2) ⊙ 3=31. 2 ⊙ 3=−4, so 3 ⊙ (2 ⊙ 3)=19. The requested difference is 31−19=12. Method choice: Evaluate each parenthesized expression separately; a newly defined operation need not be associative.
Transfer with solution
A supplier charges $3 each for the first 3 items in an order and $2 each for every additional item. What is the total charge for 10 items? The first 3 items cost $9. The remaining 7 items cost $14. The cumulative charge is $23. Method choice: Split the quantity at the tier boundary and add the cost of each portion at its own rate.
Boundary check
A real input x is doubled, then 3 is subtracted, and the result is squared to give 121. The value immediately before squaring is positive. What is x? The positive intermediate value must be 11. Thus 2x−3=11. Undo subtraction and divide by two: x=7. Method choice: Undo the operations in reverse order, using the sign condition to select the square-root branch.
Misconception to reject
The inner and outer stages of a composite can exclude different inputs. Finds two restrictions but drops the reciprocal correction in the second input.
Let f(x)=1/(x−3). What is the sum of all real inputs x for which f(f(x)) is undefined?
- The inner function is undefined at x=3.
- The outer function also fails when 1/(x−3)=3, giving x=3+1/3.
- The two excluded inputs sum to 19/3.
Compare the other answer choices
Checks only the inner function’s undefined input.
Finds two restrictions but drops the reciprocal correction in the second input.
Checks only the outer denominator’s restriction and misses the inner undefined input.
Moves the reciprocal term with the wrong sign when solving the outer restriction.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam content