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Worked solution

Reverse a horizontal stretch, reflection and translation

The original y=f(x)y=f(x) undergoes, in order:

A:translation 4 units in the positive x-direction;

B:reflection in the y-axis;

C:stretch parallel to x by 2.

The resulting equation is y=(x−1)/(x+1)y=(x-1)/(x+1). Obtain the original equation.

[2011/SAJC/II/1(b)]

CJC · Discussion · Question 4

Answer

f(x)=(2x+9)/(2x+7)f(x)=(2x+9)/(2x+7), x≠−7/2.

Undo C: horizontal scale 1/2

y=(x−1)/(x+1)⟶y=(2x−1)/(2x+1)y=(x-1)/(x+1)\longrightarrow y=(2x-1)/(2x+1).

How did this part go?

Undo B: reflect in the y-axis

Replace x by −x: y=[2(−x)−1]/[2(−x)+1]=(−2x−1)/(−2x+1)y=[2(-x)-1]/[2(-x)+1]=(-2x-1)/(-2x+1).

How did this part go?

Undo A: translate 4 units left

Replace x by x+4: y=f(x)=[−2(x+4)−1]/[−2(x+4)+1]=(−2x−9)/(−2x−7)=(2x+9)/(2x+7)y=f(x)=[-2(x+4)-1]/[-2(x+4)+1]=(-2x-9)/(-2x-7)=(2x+9)/(2x+7). Its domain excludes x=−7/2.

How did this part go?

Common mistakes

Horizontal stretching changes every x coordinate, including the location of a previous translation.

Solution notes

All printed inverse operations and intermediate rational expressions retained. No source figure.

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