What This Question Is Really Testing

The key distinction is between a step response and a transfer function . A step response already includes the dynamics of the system and the 1/s factor contributed by the step input.

Concept Foundation

For an LTI system with zero initial conditions:

G(s) = Y(s)/U(s)

If the input is a unit step, then U(s)=1/s, so the useful shortcut is G(s)=sY(s).

Step 1 — Take the Laplace Transform of the Step Response

y(t) = 2u(t) − e^−tu(t)

Y(s) = 2/s − 1/(s + 1)

Step 2 — Combine the Two Terms

Y(s) = [2(s + 1) − s]/[s(s + 1)]

Y(s) = (s + 2)/[s(s + 1)]

Step 3 — Divide by the Unit-Step Input

U(s) = 1/s

G(s) = Y(s)/(1/s)

G(s) = sY(s)

G(s) = (s + 2)/(s + 1)

Hence the correct option is C .

Why This Method Works

The step input contributes one factor 1/s to the output transform. Removing that input factor leaves only the system transfer function.

Independent Verification / Cross-Check

Final step-response value = 2

G(0) = (0 + 2)/(0 + 1) = 2

The DC gain of the derived transfer function equals the final value of the stable unit-step response, so the answer is consistent.

Concept Extension

• Impulse-response transform: H(s)=G(s).
• Unit-step-response transform: Y(s)=G(s)/s.

Exam Strategy

When a unit-step response is given, first transform it and then use G(s)=sY(s) under zero initial conditions. Use the final value as a fast DC-gain check.

Quick Trick

Unit-step response given? Transform it and multiply by s.

Why Other Options Are Wrong

A and B do not reproduce the given step response. D has G(0)=1/2, whereas the given step response settles to 2.

Common Mistake

Using Y(s) directly as G(s) and forgetting that the unit-step input itself has U(s)=1/s.

What This Question Is Really Testing

The key distinction is between a step response and a transfer function . A step response already includes the dynamics of the system and the 1/s factor contributed by the step input.

Concept Foundation

For an LTI system with zero initial conditions:

G(s) = Y(s)/U(s)

If the input is a unit step, then U(s)=1/s, so the useful shortcut is G(s)=sY(s).

Step 1 — Take the Laplace Transform of the Step Response

y(t) = 2u(t) − e^−tu(t)

Y(s) = 2/s − 1/(s + 1)

Step 2 — Combine the Two Terms

Y(s) = [2(s + 1) − s]/[s(s + 1)]

Y(s) = (s + 2)/[s(s + 1)]

Step 3 — Divide by the Unit-Step Input

U(s) = 1/s

G(s) = Y(s)/(1/s)

G(s) = sY(s)

G(s) = (s + 2)/(s + 1)

Hence the correct option is C .

Why This Method Works

The step input contributes one factor 1/s to the output transform. Removing that input factor leaves only the system transfer function.

Independent Verification / Cross-Check

Final step-response value = 2

G(0) = (0 + 2)/(0 + 1) = 2

The DC gain of the derived transfer function equals the final value of the stable unit-step response, so the answer is consistent.

Concept Extension

• Impulse-response transform: H(s)=G(s).
• Unit-step-response transform: Y(s)=G(s)/s.

Exam Strategy

When a unit-step response is given, first transform it and then use G(s)=sY(s) under zero initial conditions. Use the final value as a fast DC-gain check.

Quick Trick

Unit-step response given? Transform it and multiply by s.

Why Other Options Are Wrong

A and B do not reproduce the given step response. D has G(0)=1/2, whereas the given step response settles to 2.

Common Mistake

Using Y(s) directly as G(s) and forgetting that the unit-step input itself has U(s)=1/s.

What This Question Is Really Testing

The key distinction is between a step response and a transfer function . A step response already includes the dynamics of the system and the 1/s factor contributed by the step input.

Concept Foundation

For an LTI system with zero initial conditions:

G(s) = Y(s)/U(s)

If the input is a unit step, then U(s)=1/s, so the useful shortcut is G(s)=sY(s).

Step 1 — Take the Laplace Transform of the Step Response

y(t) = 2u(t) − e^−tu(t)

Y(s) = 2/s − 1/(s + 1)

Step 2 — Combine the Two Terms

Y(s) = [2(s + 1) − s]/[s(s + 1)]

Y(s) = (s + 2)/[s(s + 1)]

Step 3 — Divide by the Unit-Step Input

U(s) = 1/s

G(s) = Y(s)/(1/s)

G(s) = sY(s)

G(s) = (s + 2)/(s + 1)

Hence the correct option is C .

Why This Method Works

The step input contributes one factor 1/s to the output transform. Removing that input factor leaves only the system transfer function.

Independent Verification / Cross-Check

Final step-response value = 2

G(0) = (0 + 2)/(0 + 1) = 2

The DC gain of the derived transfer function equals the final value of the stable unit-step response, so the answer is consistent.

Concept Extension

• Impulse-response transform: H(s)=G(s).
• Unit-step-response transform: Y(s)=G(s)/s.

Exam Strategy

When a unit-step response is given, first transform it and then use G(s)=sY(s) under zero initial conditions. Use the final value as a fast DC-gain check.

Quick Trick

Unit-step response given? Transform it and multiply by s.

Why Other Options Are Wrong

A and B do not reproduce the given step response. D has G(0)=1/2, whereas the given step response settles to 2.

Common Mistake

Using Y(s) directly as G(s) and forgetting that the unit-step input itself has U(s)=1/s.