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Worked solution

Distinguish a squared input from a squared output

The function ff is defined by f(x)=x2−5xf(x)=x^2-5x, x∈Rx\in\mathbb R. Find (a) f(x+2)f(x+2), (b) f(x2)f(x^2), (c) [f(x)]2[f(x)]^2.

CJC · Functions · Self-Practice 2

(a) Shift the input

Answer

x2−x−6x^2-x-6.

Independently expand f(x+2)=(x+2)2−5(x+2)=x2+4x+4−5x−10=x2−x−6f(x+2)=(x+2)^2-5(x+2)=x^2+4x+4-5x-10=x^2-x-6, matching the printed answer.

How did this part go?

(b) Square the input first

Answer

x4−5x2x^4-5x^2.

f(x2)=(x2)2−5x2=x4−5x2f(x^2)=(x^2)^2-5x^2=x^4-5x^2. This substitutes x² into the rule.

How did this part go?

(c) Square the output

Answer

x4−10x3+25x2x^4-10x^3+25x^2.

[f(x)]2=(x2−5x)2=x4−10x3+25x2[f(x)]^2=(x^2-5x)^2=x^4-10x^3+25x^2. The cross term belongs here; this is a different operation from f(x²).

How did this part go?

Solution notes

All three final polynomials are printed in both copies. Expanded derivations are independent supplements.