Find \(g \circ f\) and \(f \circ g\) for \(f(x)=x^2+2x-3\) and \(g(x)=3x-4\)
📺 Video Explanation
📝 Question
Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as:
\[ f(x)=x^2+2x-3,\qquad g(x)=3x-4 \]
Find:
- \((g\circ f)(x)\)
- \((f\circ g)(x)\)
✅ Solution
🔹 Find \((g\circ f)(x)\)
By definition:
\[ (g\circ f)(x)=g(f(x)) \]
Substitute \(f(x)=x^2+2x-3\):
\[ g(f(x))=g(x^2+2x-3) \]
Since:
\[ g(x)=3x-4 \]
So:
\[ g(x^2+2x-3)=3(x^2+2x-3)-4 \]
Simplify:
\[ =3x^2+6x-9-4 \]
\[ (g\circ f)(x)=3x^2+6x-13 \]
🔹 Find \((f\circ g)(x)\)
By definition:
\[ (f\circ g)(x)=f(g(x)) \]
Substitute \(g(x)=3x-4\):
\[ f(3x-4) \]
Since:
\[ f(x)=x^2+2x-3 \]
So:
\[ f(3x-4)=(3x-4)^2+2(3x-4)-3 \]
Expand:
\[ (3x-4)^2=9x^2-24x+16 \]
Thus:
\[ f(3x-4)=9x^2-24x+16+6x-8-3 \]
\[ =9x^2-18x+5 \]
\[ (f\circ g)(x)=9x^2-18x+5 \]
🎯 Final Answer
\[ \boxed{(g\circ f)(x)=3x^2+6x-13} \]
\[ \boxed{(f\circ g)(x)=9x^2-18x+5} \]
🚀 Exam Shortcut
- \(g\circ f\): substitute whole \(f(x)\) into \(g(x)\)
- \(f\circ g\): substitute whole \(g(x)\) into \(f(x)\)
- Expand brackets carefully and combine like terms