2F - Composite functions
2F - Content video: Composite functions
This video covers the theory and several examples relating to composite functions.
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■ Composite functions contain multiple functions within a single function.
The composition of \(f\) with \(g\) can be expressed as \(f \circ g\) or as \(f(g(x))\). In this case \(g\) is applied first followed by the function \(f\).
■ For a composite function \(f(g(x))\) to exist, the range of the inner function, \(g(x)\), must be a subset or equal to the domain of the outer function, \(f(x)\). In general: \[Range(inner)\subseteq Domain(outer)\]
- The domain of the inner function can be restricted to ensure the composite function does exist.
- The domain of the composite function \(f(g(x))\) is dependent on the domain of the inner function: \[domain(f(g(x)))=domain(g(x))\]
- The range of the composite function \(f(g(x))\) is dependent on the range of the outer function. This is a bit harder to identify as it may also be affected by the domain of the inner function too!
2F - Example 4: Composite functions (VCAA, 2010)
[VCAA, 2010 Exam 1 Question 3]
Let \(f:R^{+}\rightarrow R\), where \(f(x)=\frac{1}{x^2}\).
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2F - Example 4: Video solution
2F - Example 4: Practice
Question 1: ABC Question 2: ABC 2F - Example 4: Solutions
Question 1: ABC Question 2: ABC |
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Other Resoures
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VCAA Mathematical Methods 2006 Exam 1 - Question 1
Let \(f(x)=x^2+1\) and \(g(x)=2x+1\). Write down the rule of \(f(g(x))\). |
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VCAA Mathematical Methods 2008 Exam 1 - Question 10
Let \(f:R\rightarrow R\), \(f(x)=e^{2x}-1\).
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VCAA Mathematical Methods 2010 Exam 1 - Question 3
Let \(f:R^{+}\rightarrow R\), where \(f(x)=\frac{1}{x^2}\).
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VCAA Mathematical Methods 2011 Exam 1 - Question 4
If the function \(f\) has the rule \(f(x)=\sqrt{x^2-9}\) and the function \(g\) has the rule \(g(x)=x+5\)
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VCAA Mathematical Methods 2016 SH Exam 1 - Question 5
Let \(f:(0,\infty)\rightarrow R\), where \(f(x)=log_{e}(x)\) and \(g:R\rightarrow R\), where \(g(x)=x^2+1\).
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VCAA Mathematical Methods 2017 NH Exam 1 - Question 7
Let \(f:R\rightarrow R\), where \(f(x)=2x^3+1\), and let \(g:R\rightarrow R\), where \(g(x)=4-2x\).
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VCAA Mathematical Methods 2017 SH Exam 1 - Question 7
Let \(f:[0,\infty)\rightarrow R\), \(f(x)=\sqrt{x+1}\).
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VCAA Mathematical Methods 2018 NH Exam 1 - Question 2
Let \(f(x)=-x^2+x+4\) and \(g(x)=x^2-2\).
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