Mathematics

Mathematics

Problem

Define f:(−∞,−1)∪(−1,∞)→Rf:\left(-\infty,-1\right)\cup(-1,\infty)\to\mathbb{R} by f(x)=2x−3x+1f(x)=\dfrac{2x-3}{x+1}. Which option correctly gives f−1(x)f^{-1}(x) together with its domain? a) f−1(x)=x+3 x−2 ,  domain R∖{2}f^{-1}(x)=\dfrac{x+3}{\,x-2\,},\; \text{domain }\mathbb{R}\setminus\{2\} b) f−1(x)=−x+3 x+2 ,  domain R∖{−2}f^{-1}(x)=-\dfrac{x+3}{\,x+2\,},\; \text{domain }\mathbb{R}\setminus\{-2\} c) f−1(x)=−x+3 x−2 ,  domain R∖{2}f^{-1}(x)=-\dfrac{x+3}{\,x-2\,},\; \text{domain }\mathbb{R}\setminus\{2\} d) f−1(x)=2x−3 x+1 ,  domain R∖{−1}f^{-1}(x)=\dfrac{2x-3}{\,x+1\,},\; \text{domain }\mathbb{R}\setminus\{-1\}
Intuitive explanation

To find an inverse function, you need to swap the roles of input and output: set y=f(x)y = f(x), then solve for xx in terms of yy. The key insight is that the domain of f−1f^{-1} equals the range of ff—so after you algebraically find the inverse formula, determine which yy-values the original function can actually produce. This tells you where the inverse is defined.

Intuitive explanation

To find an inverse function, you need to swap the roles of input and output: set y=f(x)y = f(x), then solve for xx in terms of yy. The key insight is that the domain of f−1f^{-1} equals the range of ff—so after you algebraically find the inverse formula, determine which yy-values the original function can actually produce. This tells you where the inverse is defined.

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