Find \(f \circ g\) and \(g \circ f\) for Constant Function \(f(x)=c\) and \(g(x)=\sin x^2\)
📺 Video Explanation
📝 Question
Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as:
\[ f(x)=c,\quad c\in\mathbb{R} \]
and
\[ g(x)=\sin x^2 \]
Find:
- \((f\circ g)(x)\)
- \((g\circ f)(x)\)
✅ Solution
🔹 Find \((f\circ g)(x)\)
By definition:
\[ (f\circ g)(x)=f(g(x)) \]
Substitute:
\[ (f\circ g)(x)=f(\sin x^2) \]
Since \(f\) is a constant function:
\[ f(t)=c \quad \text{for every } t \]
Therefore:
\[ \boxed{(f\circ g)(x)=c} \]
🔹 Find \((g\circ f)(x)\)
By definition:
\[ (g\circ f)(x)=g(f(x)) \]
Substitute:
\[ (g\circ f)(x)=g(c) \]
Since:
\[ g(x)=\sin x^2 \]
So:
\[ g(c)=\sin(c^2) \]
Therefore:
\[ \boxed{(g\circ f)(x)=\sin(c^2)} \]
This is also a constant function.
🎯 Final Answer
\[ \boxed{(f\circ g)(x)=c} \]
\[ \boxed{(g\circ f)(x)=\sin(c^2)} \]
Hence, both compositions are constant functions.
🚀 Exam Shortcut
- Constant function always gives same output
- So \(f\circ g\) stays constant
- For \(g\circ f\), just put constant inside \(g\)