E3.3 - Composite functions (Extended)

E3.3 - Composite functions (Extended)

Start with 4. Does “add 3, then double” give the same result as “double, then add 3”? Combining two function rules makes their order visible. You will interpret composite notation, substitute a whole expression into another rule, and simplify composites that contain fractions.

Order of composition

Let f(x)=x+3f(x)=x+3 and g(x)=2xg(x)=2x. Starting at 4, applying ff then gg gives 4↦7↦144\mapsto7\mapsto14. Reversing the order gives 4↦8↦114\mapsto8\mapsto11. The same two operations can produce different results.

Composite function

A composite function applies one function to the output of another. Cambridge notation gf(x)gf(x) means g(f(x))g(f(x)): apply the function nearest the input first. It does not mean g(x)×f(x)g(x)\times f(x).

In fg(x)=f(g(x))fg(x)=f(g(x)), gg acts first. Think of the nested brackets as instructions: calculate the inside, then the outside. You already used this idea to check that an inverse undoes a function.

Worked example 1: evaluating in the stated order

Question. For f(x)=x+3f(x)=x+3 and g(x)=2xg(x)=2x, find gf(4)gf(4).

Plan. Evaluate f(4)f(4), then use that output as the input of gg.

f(4)=4+3=7inside function firstgf(4)=g(7)=2(7)=14outside function next\begin{aligned} f(4)&=4+3=7 &&\text{inside function first}\\ gf(4)&=g(7)=2(7)=14 &&\text{outside function next} \end{aligned}

Answer. 1414.

Check. Following the instruction in words, add 3 to 4 and then double: 2(4+3)=142(4+3)=14.

Keep the rules and input fixed, but reverse only the order.

Worked example 2: reversing the order

Question. With the same rules, find fg(4)fg(4).

Plan. Evaluate g(4)g(4), then apply ff.

g(4)=2(4)=8fg(4)=f(8)=8+3=11\begin{aligned} g(4)&=2(4)=8\\ fg(4)&=f(8)=8+3=11 \end{aligned}

Answer. 1111, so fg(4)≠gf(4)fg(4)\ne gf(4).

Check. Doubling 4 and then adding 3 gives 11. The difference from 14 is explained by the added 3 being doubled only in the first order.

Writing an intermediate value helps you keep the direction straight, especially when the two function names are close together.

Watch out

The wrong product g(4)f(4)=8×7=56g(4)f(4)=8\times7=56 does not evaluate either composite. Composition feeds one output into another function.

Follow the same notation with fresh functions and an input that produces a negative intermediate result.

Forming composites

To find a rule for every input, replace each input in the outside function by the whole inside expression. Brackets preserve the operations performed on that input. For example, if the outside rule squares its input, it must square the complete inside expression.

Worked example 3: a linear rule inside a square

Question. Let f(x)=2x−1f(x)=2x-1 and g(x)=x2+4g(x)=x^2+4. Find gf(x)gf(x), giving your answer in expanded form.

Plan. Put 2x−12x-1 into every input position of gg, then expand.

gf(x)=g(2x−1)=(2x−1)2+4square the whole input=4x2−4x+1+4expand the square=4x2−4x+5collect constants\begin{aligned} gf(x)&=g(2x-1)\\ &=(2x-1)^2+4 &&\text{square the whole input}\\ &=4x^2-4x+1+4 &&\text{expand the square}\\ &=4x^2-4x+5 &&\text{collect constants} \end{aligned}

Answer. gf(x)=4x2−4x+5gf(x)=4x^2-4x+5.

Check. At x=2x=2, the separate rules give f(2)=3f(2)=3, then g(3)=13g(3)=13. The expanded rule gives 16−8+5=1316-8+5=13.

Reverse only the order. The input expression is now squared before the outer doubling and subtraction.

Worked example 4: a square inside a linear rule

Question. With the same ff and gg, find fg(x)fg(x) in expanded form.

Plan. Put x2+4x^2+4 into the input position of ff.

fg(x)=f(x2+4)=2(x2+4)−1double the whole input=2x2+8−1=2x2+7\begin{aligned} fg(x)&=f(x^2+4)\\ &=2(x^2+4)-1 &&\text{double the whole input}\\ &=2x^2+8-1\\ &=2x^2+7 \end{aligned}

Answer. fg(x)=2x2+7fg(x)=2x^2+7.

Check. At x=2x=2, g(2)=8g(2)=8, then f(8)=15f(8)=15. The new rule gives 2(2)2+7=152(2)^2+7=15, different from gf(2)gf(2).

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