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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