Problem #372
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372
The function \(f\) is defined by \(f: x \mapsto x^2 - 3x\) for \(x \in \mathbb{R}\). Find the set of values of \(x\) for which \(f(x) > 4\).
Solution
To find the set of values of \(x\) for which \(f(x) > 4\), we start with the inequality:
\(x^2 - 3x > 4\)
Rearrange the inequality:
\(x^2 - 3x - 4 > 0\)
Factor the quadratic expression:
\((x - 4)(x + 1) > 0\)
Determine the critical points by setting each factor to zero:
\(x - 4 = 0 \Rightarrow x = 4\)
\(x + 1 = 0 \Rightarrow x = -1\)
Test intervals around the critical points:
- For \(x < -1\), choose \(x = -2\): \((x - 4)(x + 1) = (-2 - 4)(-2 + 1) = 6 > 0\)
- For \(-1 < x < 4\), choose \(x = 0\): \((x - 4)(x + 1) = (0 - 4)(0 + 1) = -4 < 0\)
- For \(x > 4\), choose \(x = 5\): \((x - 4)(x + 1) = (5 - 4)(5 + 1) = 6 > 0\)
Thus, the solution is \(x < -1\) or \(x > 4\).