Maths · The composition of functions
Let =\mathbf{2}^{100} \boldsymbol{x}+\mathbf{1} \) =\boldsymbol{3}^{100} \boldsy
Let \( \boldsymbol{f}(\boldsymbol{x})=\mathbf{2}^{100} \boldsymbol{x}+\mathbf{1} \) \( \boldsymbol{g}(\boldsymbol{x})=\boldsymbol{3}^{100} \boldsymbol{x}+\mathbf{1} \) Then the set of real numbers x such that \( \boldsymbol{f}(\boldsymbol{g}(\boldsymbol{x}))=\boldsymbol{x} \) is
- A. Empty
- B. A singleton
- C. A finite se with more than one element
- D. Infinite
Step-by-step solution
Compute f(g(x)): f(g(x)) = 2^{100}(3^{100}x + 1) + 1 = 6^{100}x + 2^{100} + 1. Set equal to x: 6^{100}x + 2^{100} + 1 = x → (6^{100} - 1)x = -(2^{100} + 1) → x = -(2^{100} + 1)/(6^{100} - 1), which is a unique real number. Thus the set is a singleton.
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