Consider a function $g$ defined as $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$. Sketch the grap

Question

Consider a function $g$ defined as $g:x\mapsto (x+2)^2-3,\ x\in\mathbb{R}$. Sketch the graph of $g$ and hence explain why $g$ cannot have an inverse. If $g:x\mapsto (x+2)^2-3,\ x\ge q$ has an inverse, write down the smallest value $q$ can take. For this value of $q$, find the function $g^{-1}$. Sketch the graphs of $g$ and $g^{-1}$ on the same axes, clearly showing the relationship between them. Write down the range of $g^{-1}$. Find the exact solution of the equation $g (x) =g^{-1}(x)$.

Accepted Answer

(a) $g$ is not one-to-one. (b) $q=-2$. (c) $g^{-1}:x\mapsto-2+\sqrt{x+3}$, $x\ge-3$. (e) $[-2,\infty)$. (f) $x=\frac{-3+\sqrt5}{2}$.

Functions - Sketching a Function and Its Inverse

Functions - Sketching a Function and Its Inverse

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