Problem

For \(R(x)~=~[(x~-~3)(x~+~2)]/(x~-~3)\) with \(x~\ge~0\), cancel valid factors, retain every original restriction, state the reduced model, locate the hole, identify any remaining vertical asymptote, and compare the exclusions with the contextual domain.

Big Picture

What this problem is really about

Canceling a common factor simplifies the rule but does not restore an input that the original denominator excluded. Carry that restriction into the reduced graph, evaluate the simplified expression at the missing input to locate the hole, and then decide whether any uncanceled denominator factor remains to create a vertical asymptote. Finally compare the exclusion with the contextual domain.

Answer choices

Choose your path.

Preview only

Cancellation: remove x−3 but retain x≠3. Reduced model: R=x+2 on x≥0, x≠3. Hole: (3,0). Remaining vertical asymptote: none; the excluded input lies inside the contextual domain.

Cancellation: remove x−3 but retain x≠3. Reduced model: R=x+2 on x≥0, x≠3. Hole: (3,5). Remaining vertical asymptote: none; the excluded input lies inside the contextual domain.

Cancellation: do not remove x−3. Reduced model: unchanged on x≥0, x≠3. Hole: none. Remaining vertical asymptote: x=3 inside the contextual domain.

Cancellation: remove x−3 and restore x=3. Reduced model: R=x+2 on all x≥0. Hole: none. Remaining vertical asymptote: none; every contextual input is included.

The choices are visible. Checking your work and seeing which answer is correct happen inside Gozunta.

For \(R(x)~=~[(x~-~3)(x~+~2)]/(x~-~3)\) with \(x~\ge~0\), cancel valid factors, retain every original restriction, state the reduced model, locate the hole, identify any remaining vertical asymptote, and compare the exclusions with the contextual domain.

What this problem is really about

Turn the preview into practice.

Gozunta lets you study this problem—and more than 10,000 others.