Let a bijective function g : R → R be defined as g(x) = \left{\right. x + \left(\alpha\right)^{2} + 2 , \textrm{ }\textrm{ }\textrm{ } x \leq 2 \\ 7 + αx , \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } x > 2. If graph of y = f(x) is reflection of graph of y = g(x) w.r.t. line y = x, then find f(11).

Engineering

Mathematics

Inverse of a Function

Question

Let a bijective function g : R → R be defined as g(x) = ${\begin{cases} x + α^{2} + 2, x \leq 2 \\ 7 + αx, x > 2 \end{cases}$ . If graph of y = f(x) is reflection of graph of y = g(x) w.r.t. line y = x, then find f(11).

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For g(x) to be bijective
Clearly   α > 0
and   4 + α² = 7 + 2α
⇒    α² – 2α – 3 = 0
⇒    (α – 3)(α + 1) = 0

$∴ g (x) = {\begin{cases} x + 11, x \leq 2 \\ 3 x + 7, x > 2 \end{cases}$
f (x) is inverse of g(x)
and f (11) = g^–1(11)
and g(x) = 11 ⇒ x = 0 ⇒ g^–1(11) = 0