The decision and its conditions
An inverse reverses a function’s input-output pairing, so the original range becomes the inverse domain. A quadratic needs a one-to-one domain restriction before its inverse is a function. Choose the square-root branch consistent with that restriction, not whichever sign is convenient. A rational inverse requires solving a linear equation in the original input after cross multiplication, while keeping its own forbidden output visible. Composition and quotients impose simultaneous domain conditions: an inner output must be allowed by the outer rule, and every original denominator must remain nonzero.
Worked example
f(x)=(x−3)²+5 has domain x≥3. What is f⁻¹(9)? Set (x−3)²=4. The domain requires x−3≥0, so x−3=2. The inverse output is x=5; the other square-root branch is outside the original domain.
A contrasting decision
f(x)=(x+6)/(x−3). Find f⁻¹(2). Solve 2=(x+6)/(x−3) with x≠3. Cross multiplication gives 2x−6=x+6. Thus x=12; it is allowed and yields output two.
Diagnose a competing method
The first example’s tempting answer 1 follows this mistake: Use the forbidden negative square-root branch. Use the branch specified by the original function’s domain.
Transfer before revealing
For f(u)=√u and g(x)=7−x, how many integer inputs x in [3,11] belong to the domain of f(g(x))? Pause and identify the original restrictions. Solution: The outer square root requires 7−x≥0, hence x≤7. Intersect with the given input window: 3≤x≤7. There are five integers, including both endpoints.
Check the boundary of the method
A rational inverse requires solving a linear equation in the original input after cross multiplication, while keeping its own forbidden output visible. An inverse value asks for the original input that produces the specified output. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
f(x)=(x−3)²+5 has domain x≥3. What is f⁻¹(9)?
- Set (x−3)²=4.
- The domain requires x−3≥0, so x−3=2.
- The inverse output is x=5; the other square-root branch is outside the original domain.
Compare the other answer choices
Use the forbidden negative square-root branch.
Forget to restore the horizontal shift.
Add the vertical shift instead of the horizontal shift.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Show substituted work, state the required domain or geometric condition, and refute a competing method. This extension is not automatically scored.
This draft stays on this page only. Use paper if you want to keep your working.
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.
CLEP College Algebra exam overview