Which of the pairs of functions below are not inverse functions?
- \( \begin {cases}f(x)=5x+3 &\ \ x\in R \\ g(x)=\dfrac{x-3}{5} &\ \ x \in R \\ \end{cases}\)
- \( \begin {cases}f(x)=\frac{2}{3}x+2 &\ \ x\in R \\ g(x)=\frac{3}{2}x-3 &\ \ x \in R \\ \end{cases}\)
- \( \begin {cases}f(x)=x^2 &\ \ x<0 \\ g(x)=\sqrt{x} &\ \ x >0 \\ \end{cases}\)
- \( \begin {cases}f(x)=\dfrac{1}{x} &\ \ x\neq 0 \\ g(x)=\dfrac{1}{x} &\ \ x \neq 0 \\ \end{cases}\)
- \( \begin {cases}f(x)=\log_e(x)+1 &\ \ x>0 \\ g(x)=e^{x-1} &\ \ x \in R \\ \end{cases}\)
Show Worked Solution
\(\text{Graphically, it can be seen that } f(x)=x^2\ \text{and }g(x)=\sqrt{x}\ \text{are not}\)
\(\text{inverse functions as }g(x)\ \text{is not a reflection of }f(x)\ \text{in the line }y=x.\)
\(g(x)\ \text{should be the curve drawn as } h(x)=-\sqrt{x}\ \text{below}.\)
\(\Rightarrow C\)
♦ Mean mark 47%.
