Decision and conditions
Treat a function as an instruction with a permitted input and a selected branch. Cancel common factors only after retaining excluded inputs; use the branch whose condition contains the input.
Worked model
For x ≠ 3, let f(x) = (x² − 9)/(x − 3). What is f(4)? The numerator is (x − 3)(x + 3). At the permitted input 4, cancel the common nonzero factor. f(4) = 4+3 = 7. Before calculating: Factor the numerator, keeping the original excluded input.
Contrast: a different decision
Let f(x) = x² + 1 and g(x) = 2x − 3. What is f(g(4))? g(4) = 5. f(g(4)) = (5)² + 1. The value is 26. Before calculating: Evaluate the inner function first, then use its output as the new input.
Transfer with solution
Define f(x)=x+3 when x<0 and f(x)=x² when x≥0. What is f(−4) + f(3)? The negative input uses x+3, giving −1. The nonnegative input uses x², giving 9. The sum is 8. Before calculating: Choose a branch independently for each input before adding outputs.
Check the boundary
Composition changes the input before the outer function acts; multiplication of outputs is different. The function f(x) = 3x − 3 is invertible. What is f⁻¹(9)? Solve 3x − 3 = 9. Add 3 to obtain 3x = 12. The input is 4.
For x ≠ 3, let f(x) = (x² − 9)/(x − 3). What is f(4)?
- The numerator is (x − 3)(x + 3).
- At the permitted input 4, cancel the common nonzero factor.
- f(4) = 4+3 = 7.
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.
GRE Quantitative Reasoning overview