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    5. Mathematics Applications & Interpretation (Math AI)
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    The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, x \in \mathbb{R}$.

    Question
    HLPaper 2

    The function f(x)=ln⁡(1x−2)f(x)=\ln\left(\frac{1}{x-2}\right)f(x)=ln(x−21​) is defined for x>2,x∈Rx>2, x \in \mathbb{R}x>2,x∈R.

    1.

    Find an expression for f−1(x)f^{-1}(x)f−1(x). You are not required to state a domain.

    [4]
    Verified
    Solution
    • Set y=ln⁡(1x−2)y=\ln(\frac{1}{x-2})y=ln(x−21​) and solve for xxx such that ey=1x−2e^y=\frac{1}{x-2}ey=x−21​ M1 A1
    • Hence x−2=1ey⟹x=1ey+2x-2=\frac{1}{e^y} \Longrightarrow x=\frac{1}{e^y}+2x−2=ey1​⟹x=ey1​+2 M1
    • So f−1(x)=1ex+2f^{-1}(x)=\frac{1}{e^x}+2f−1(x)=ex1​+2 A1
    2.

    Solve f(x)=f−1(x)f(x)=f^{-1}(x)f(x)=f−1(x).

    [1]
    Verified
    Solution
    • f(x)=f−1(x)⟹ln⁡(1x−2)=1ex+2f(x)=f^{-1}(x)\Longrightarrow\ln(\frac{1}{x-2})=\frac{1}{e^x}+2f(x)=f−1(x)⟹ln(x−21​)=ex1​+2 using GDC
    • x≈2.12x\approx2.12x≈2.12 A1

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    Related topics

    AHL 1.9—Log lawsAHL 2.7—Composite functions, finding inverse function incl domain restrictionSL 1.5—Exponents and Logarithms
    Mathematics Applications & Interpretation (Math AI)
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