Question

Students are asked to graph the inverse, f–1(x) of an absolute value function, f(x) after restricting its domain. The directions also ask that the graph include a dashed line that can be used to verify that the functions are inverses of each other.

Which student’s work is correct?

Answer 1

I don’t see the students’ graphs or equations — could you upload or describe each student’s work (their f(x), the restricted domain, and their graphed f⁻¹(x) and dashed line)? With that I can say which is correct.

Meanwhile, here is a quick checklist you (or I) can use to decide which student is correct:

1. Is the original absolute-value function made one-to-one by the stated domain restriction?
- An unrestricted |x − h| is not one-to-one; you must restrict to one branch (x ≥ h or x ≤ h) so it’s monotonic.

2. Does the inverse match algebraically?
- Start with y = f(x) (using the restricted branch so the absolute value becomes ±(x − h)).
- Solve for x in terms of y, then swap x and y. The result is f⁻¹(x).
- Check the domain of f⁻¹ equals the range of the restricted f.

3. Is the graphed f⁻¹ the reflection of f across the line y = x?
- The dashed verification line should be y = x.
- Graphically, every point (a, b) on f should correspond to (b, a) on f⁻¹.
- Pick a few easy points on f and make sure their coordinates are swapped on f⁻¹.

4. Does the inverse’s domain and range make sense?
- Domain(f⁻¹) = Range(f on the restricted domain).
- Range(f⁻¹) = Domain(f on the restricted domain).

Common mistakes to look for
- Not restricting the domain (or restricting the wrong branch).
- Solving for the inverse but forgetting to adjust the domain of the inverse.
- Drawing the reflection incorrectly (e.g., reflecting across the wrong line, or shifting instead of reflecting).

If you post the students’ graphs or equations I’ll check them step-by-step and tell you which one is correct.