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Math Help - Solving an inequality

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    Solving an inequality

    Function f: R \rightarrow R is a periodic function with period T=6. For x \in <-3,3>, the formula of this function is f(x)=9-x^2. Solve the inequality f(x) \le 5.
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    Quote Originally Posted by bobb12 View Post
    Function f: R \rightarrow R is a periodic function with period T=6. For x \in <-3,3>, the formula of this function is f(x)=9-x^2. Solve the inequality f(x) \le 5.
    Note that \mathbb{R} gives \mathbb{R}

    First solve the inequality for x \in [-3,3] this gives

    9-x^2 \le 5 \iff 4-x^2 \le 0 \iff (2-x)(2+x) \le 0 so the solution is

    [-3,-2] \cup [2,3]

    Since the function is periodic we know that f(x)=f(x+6) So the inequality must must hold for when we shift right or left by 6 this gives

    [-3+6n,-2+6n] \cup [2 +6n,3 +6n]=[2 +6n,4 +6n], n \in \mathbb{Z}
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