Composite functions
A composite function applies one function to the output of another. The inner rule is applied first, and the resulting outputs must be allowed inputs for the outer rule
\[\left(f\circ g\right)(x)=f\left(g(x)\right)\]
- Read the order from the input The function next to x is the inner function and receives the input first \(\displaystyle \left(f\circ g\right)(x)=f\left(g(x)\right)\)
- Substitute the complete inner rule Replace every occurrence of the outer input with the full inner expression \(\displaystyle f(u)=u^2+1,\quad g(x)=x-2\Rightarrow f\left(g(x)\right)=\left(x-2\right)^2+1\)
- Check the range-domain condition Every output of the inner function must be in the domain of the outer function \(\displaystyle \operatorname{Ran}(g)=[1,5]\subseteq\operatorname{Dom}(f)=[0,5]\)
- Restrict the inner domain when needed Here, start with g(x)=2x on R and f with domain [2,5]. Restrict g to inputs whose outputs are allowed by f before forming the composite \(\displaystyle g(x)=2x,\quad 2\le2x\le5\Rightarrow\operatorname{Dom}(f\circ g)=\left[1,\frac{5}{2}\right]\)
Worked example
For the functions shown, find the composite rule, its domain and its value at the stated input
\[f(x)=2x+1,\quad g(x)=x^2-3,\quad -1\le x\le3,\quad x=2,\quad left(fcirc g
ight)(x)\]
- \(\displaystyle \left(f\circ g\right)(x)=f\left(g(x)\right)=2\left(x^2-3\right)+1\)Apply g first and substitute its complete rule into f
- \(\displaystyle \left(f\circ g\right)(x)=2x^2-6+1=2x^2-5\)Expand and collect the constant terms
- \(\displaystyle \operatorname{Dom}(f\circ g)=[-1,3]\)f accepts every real input, so the composite keeps the stated domain of g
- \(\displaystyle \left(f\circ g\right)(2)=2(2)^2-5=8-5=3\)Substitute x=2 into the simplified composite rule
Common mistakes
Composition order is reversed
✕\(\displaystyle \left(f\circ g\right)(x)=g\left(f(x)\right)\)✓\(\displaystyle \left(f\circ g\right)(x)=f\left(g(x)\right)\)
The function nearest to x is applied first, so the composition is read from right to leftOnly one occurrence of the inner rule is substituted
✕\(\displaystyle f(u)=u^2+2u,\quad g(x)=x+1\Rightarrow f\left(g(x)\right)=\left(x+1\right)^2+2x\)✓\(\displaystyle f(u)=u^2+2u,\quad g(x)=x+1\Rightarrow f\left(g(x)\right)=\left(x+1\right)^2+2\left(x+1\right)\)
Every occurrence of the outer input must be replaced by the complete inner ruleThe outer domain is used as the composite domain
✕\(\displaystyle \operatorname{Dom}(f\circ g)=\operatorname{Dom}(f)\)✓\(\displaystyle \operatorname{Dom}(f\circ g)=\operatorname{Dom}(g)\text{ after checking }\operatorname{Ran}(g)\subseteq\operatorname{Dom}(f)\)
The composite starts with inner inputs and removes any that produce an invalid outer inputDifficulty levels
introForm both orders of a polynomial or linear composite without a domain complication
standardCheck existence or select the domain and range for composites involving root or reciprocal rules
challengeSolve an inequality to restrict the inner domain before forming the composite rule
examComplete a compact item involving existence, rule coefficients, domain and range
Questions about this topic
How does composition differ from multiplying function values?+
In a composition, the inner function's output becomes the outer function's input. A product multiplies the two function outputs
Why is the composition order read from right to left?+
The rightmost function receives the original input first, and the next function receives the output produced by it
How is the domain of a composite found?+
Start with the inner function's domain and keep only inputs whose inner outputs lie in the outer function's domain
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 3: Functions, transformations and inverses