Apply the inner function first and keep the whole substituted expression.
Extended · Syllabus E2.13
Notes and key methods
Composition is successive application, rather than multiplication. Here fg means f∘g, so fg(x)=f(g(x)); gf applies f first. Brackets are essential when the outer rule includes a square or a reciprocal.
In general the two orders give different rules. One matching value would not prove that two functions are identical; compare their complete expressions. Domains and ranges of composite functions are not required for this syllabus, and the rules in these exercises are defined for every real input.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Composite functions and order
\(f(x)=3 x + 9\), \(g(x)=\left(x + 1\right)^{2}\) and \(\displaystyle t=2\). Both rules have domain ℝ. Use fg=f∘g and gf=g∘f.
Find and simplify fg(x).
Find and simplify gf(x).
Compare the compositions at t and explain whether they are identical functions.
Find the coefficient of \(\displaystyle x^{2}\) in f(g(x)).
Show the reasoning
(a)
\(fg(x)=3 \left(x + 1\right)^{2} + 9\) expands to \(3 x^{2} + 6 x + 12\).
Answer: \(3 x^{2} + 6 x + 12\)
(b)
\(gf(x)=\left(3 x + 10\right)^{2}\) expands to \(9 x^{2} + 60 x + 100\).
Answer: \(9 x^{2} + 60 x + 100\)
(c)
The values at this t differ. Independently, the \(\displaystyle x^{2}\) coefficients of the complete rules are 3 and 9, which are unequal.
Answer: \(fg(t)=36,\quad gf(t)=256\). They are different functions.
(d)
Apply the inner rule first and substitute its whole output into the outer rule before simplifying.
\(3\)
Answer: \(3\).
Example 2 · Composite functions and order
\(f(x)=4 x + 10\), \(g(x)=x^{2}\) and \(\displaystyle t=4\). Both rules have domain ℝ. Use fg=f∘g and gf=g∘f.
Find and simplify fg(x).
Find and simplify gf(x).
Compare the compositions at t and explain whether they are identical functions.
Find the coefficient of \(\displaystyle x^{2}\) in f(g(x)).
Show the reasoning
(a)
\(fg(x)=4 x^{2} + 10\) expands to \(4 x^{2} + 10\).
Answer: \(4 x^{2} + 10\)
(b)
\(gf(x)=\left(4 x + 10\right)^{2}\) expands to \(16 x^{2} + 80 x + 100\).
Answer: \(16 x^{2} + 80 x + 100\)
(c)
The values at this t differ. Independently, the \(\displaystyle x^{2}\) coefficients of the complete rules are 4 and 16, which are unequal.
Answer: \(fg(t)=74,\quad gf(t)=676\). They are different functions.
(d)
Apply the inner rule first and substitute its whole output into the outer rule before simplifying.
\(4\)
Answer: \(4\).
Example 3 · Composite functions and order
\(f(x)=5 x + 2\), \(g(x)=\left(x - 1\right)^{2}\) and \(\displaystyle t=6\). Both rules have domain ℝ. Use fg=f∘g and gf=g∘f.
Find and simplify fg(x).
Find and simplify gf(x).
Compare the compositions at t and explain whether they are identical functions.
Find the coefficient of \(\displaystyle x^{2}\) in f(g(x)).
Show the reasoning
(a)
\(fg(x)=5 \left(x - 1\right)^{2} + 2\) expands to \(5 x^{2} - 10 x + 7\).
Answer: \(5 x^{2} - 10 x + 7\)
(b)
\(gf(x)=\left(5 x + 1\right)^{2}\) expands to \(25 x^{2} + 10 x + 1\).
Answer: \(25 x^{2} + 10 x + 1\)
(c)
The values at this t differ. Independently, the \(\displaystyle x^{2}\) coefficients of the complete rules are 5 and 25, which are unequal.
Answer: \(fg(t)=127,\quad gf(t)=961\). They are different functions.
(d)
Apply the inner rule first and substitute its whole output into the outer rule before simplifying.
\(5\)
Answer: \(5\).
Common mistakes
Keep intermediate values exact until the final step.