where \(x \in \mathbb{R}\) in each case.
- Find the solution set, \(S\), of the inequality \(f(x) \ge g(x)\).
Finding when \(f(x) \ge g(x)\) is the same as finding when \(f(x)-g(x)\ge0\).
We could expand the brackets, simplify and factorise, but instead notice that the two functions have a common factor…
So the graph of \(y=f(x)-g(x)\) is a parabola with a maximum and \(x\)-intercepts at \(x=1\) and \(x=\frac{5}{2}\). \(y\) is greater than zero between the two roots and the roots themselves are included in the interval.
\(S\) is therefore the set of values \[1\le x\le\frac{5}{2}.\]
- Sketch the graph of \(y = f(x) - g(x)\) for \(x \in S\), and state the greatest and least values of \(f(x) - g(x)\) for \(x \in S\).
We noted above that the graph of \(y=f(x)-g(x)\) is a parabola with a maximum and zeros at \(x=1\) and \(x=\frac{5}{2}\). The minimum value of \(y\) will be zero and the maximum value will occur half way between the two zeros, that is at \(x=\frac{7}{4}\). \[y_{max} = -2\left( \frac{7}{4}-1 \right)\left( \frac{7}{4}-\frac{5}{2} \right) = \frac{9}{8}\]
