1 Composite Function
정의 1 Given two functions f and g, the composite function \(f\circ g\) (also called the composition of \(f\) and \(g\)) is defined by
\[ (f\circ g)(x)=f(g(x)) \tag{1}\]
The domain of \(f\circ g\) is the set of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\). In other words, \((f\circ g)(x)\) is defined whenever both \(g(x)\) and \(f(g(x))\) are defined.
1.1 Example
If \(f(x)=x^2\) and \(g(x)=e^x\), find \(f\circ g\) and \(g\circ f\).
\[ \begin{aligned} (f\circ g)(x)&=f(g(x))=f(e^x)=(e^x)^2=e^{2x}\\ (g\circ f)(x)&=g(f(x))=g(x^2)=e^{x^2}=e^{x^2}\\ \end{aligned} \]
위의 예제처럼, \(f\circ g \ne f\circ g\).
2 Composite Function
정의 2 Given two functions f and g, the composite function \(f\circ g\) (also called the composition of \(f\) and \(g\)) is defined by
\[ (f\circ g)(x)=f(g(x)) \tag{2}\]
The domain of \(f\circ g\) is the set of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\). In other words, \((f\circ g)(x)\) is defined whenever both \(g(x)\) and \(f(g(x))\) are defined.
2.1 Example
If \(f(x)=x^2\) and \(g(x)=e^x\), find \(f\circ g\) and \(g\circ f\).
\[ \begin{aligned} (f\circ g)(x)&=f(g(x))=f(e^x)=(e^x)^2=e^{2x}\\ (g\circ f)(x)&=g(f(x))=g(x^2)=e^{x^2}=e^{x^2}\\ \end{aligned} \]
Like the example above, \(f\circ g \ne f\circ g\).
#| echo: false
#| eval: false
radius = 10
from IPython.display import display, Markdown
display(Markdown("""
The radius of the circle is {radius}.
""".format(radius = radius)))