Mathematics - Functions Question with Solution | TestHub

$f(x)=y=x^2-x+5$ $$\begin{array}{rl}& x^2-x+\frac{1}{4}-\frac{1}{4}+5=y\\ & {\left(x-\frac{1}{2}\right)}^2+\frac{19}{4}=y\\ & {\left(x-\frac{1}{2}\right)}^2=y-\frac{19}{4}\\ & x-\frac{1}{2}=\pm \sqrt{y-\frac{19}{4}}\end{array}$$ $$\begin{array}{rl}& x=\frac{1}{2}\pm \sqrt{y-\frac{19}{4}}\\ & \text{ As }x>\frac{1}{2}\\ & x=\frac{1}{2}+\sqrt{y-\frac{19}{4}}\\ & g(x)=\frac{1}{2}+\sqrt{x-\frac{19}{4}}\\ & g^{\prime }(x)=\frac{1}{2\sqrt{x-\frac{19}{4}}}\end{array}$$ $$g^{\prime }(7)=\frac{1}{2\sqrt{7-\frac{19}{4}}}=\frac{1}{2\frac{\sqrt{28-19}}{2}}=\frac{1}{3}$$