Sets, Relations & Functions Question - None

Question

Let $f:(0,\infty)\to(1,\infty)$, $f(x)=1+\frac{3}{2}\sqrt{x}$, and $g=f^{-1}$. Find the intersection point of $f$ and $g$.

Accepted Answer

Key Idea: f and f⁻^1 intersect on y=x. Solve f(x)=x.

Step 1: $x = 1+\frac{3}{2}\sqrt{x}$. Let $t=\sqrt{x}>0$.

Step 2: $2t^2-3t-2=0 \implies (2t+1)(t-2)=0 \implies t=2$

Step 3: $x=4$, intersection point is $(4,4)$

Trap: Writing the inverse first is slower. Use symmetry line y=x directly.

Key Concept: f and f⁻^1 intersect on y=x

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