Goal and prerequisites
Interpret sign intervals, choose an inverse branch and identify intersections without confusing outputs with inputs. Review quadratics sequences before starting.
Method and its conditions
For a factored quadratic, use its zeros to split the number line and test each sign interval. A square function needs a one-to-one domain restriction before inversion. For intersections, equate the two y expressions and check the resulting coordinate pairs.
Worked example: show the decisions
The inequality x(x−8)<0 holds for 0<x<8, giving seven integer solutions. The function f(x)=(x−2)² on x≥2 has inverse f⁻¹(y)=2+√y, not both square-root branches. Thus f⁻¹(9)=5.
Contrast: diagnose an incorrect approach
Solving x²=y gives two candidates unless a domain restriction selects one. For a line y=x+6 and parabola y=x², intersection inputs solve x²−x−6=0, so x=3 or−2. The corresponding outputs are 9 and 4, not the same list as the inputs.
Transfer: try a changed situation
Find the intersections of y=x+12 and y=x², and state the horizontal asymptote of y=3/(x−1)+5.
Transfer solution and interpretation
The equation factors as (x−4)(x+3)=0, giving (4,16) and (−3,9). The reciprocal term tends to zero as |x| grows, so the horizontal asymptote is y=5.
How many integers satisfy x(x−6)<0?
- The upward quadratic is negative strictly between its roots 0 and the stated positive root.
- Using the stated quantities: 6-1 = 5.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Compare the other answer choices
Count roots instead of integer solutions.
Use interval length instead of the count of interior integers.
Include both zero endpoints in a strict inequality.
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.
0580 syllabus for exams in 2025–2027