Find Linear Functions Mapping Intervals
🎥 Video Explanation
📝 Question
Find all linear functions mapping \([-1,1]\) onto \([0,2]\).
- A. \(f(x)=x+1,\; g(x)=-x+1\)
- B. \(f(x)=x-1,\; g(x)=x+1\)
- C. \(f(x)=-x-1,\; g(x)=x-1\)
- D. none of these
✅ Solution
🔹 Step 1: General Form
Let:
\[ f(x)=ax+b \] —
🔹 Step 2: Endpoint Mapping
Case 1: Increasing function
\[ f(-1)=0,\quad f(1)=2 \]
\[ -a+b=0 \Rightarrow b=a \]
\[ a+b=2 \Rightarrow a+a=2 \Rightarrow a=1,\; b=1 \]
So:
\[ f(x)=x+1 \] —
Case 2: Decreasing function
\[ f(-1)=2,\quad f(1)=0 \]
\[ -a+b=2 \]
\[ a+b=0 \Rightarrow b=-a \]
Substitute:
\[ -a-a=2 \Rightarrow -2a=2 \Rightarrow a=-1,\; b=1 \]
So:
\[ g(x)=-x+1 \] —
🔹 Final Answer
\[ \boxed{\text{Option A}} \]