Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 25–32, use the given functions f and g. 

$$\begin{gathered} \left(a\right) f\left(x\right)=0 \\ \left(b\right) g\left(x\right) = 0 \\ \left(c\right) f\left(x\right)= g\left(x\right) \\ \left(d\right) fx>0 \\ \left(e\right) g\left(x\right)\le 0 \\ \left(f\right) f\left(x\right)>g\left(x\right) \\ \left(g\right) f\left(x\right)\ge 1 \\ f\left(x\right)=x^2-4 \\ g\left(x\right)=-x^2+4 \end{gathered}$$

Step-by-Step Solution

Answer

(a) x=+/-2; (b) x=+/-2; (c) x=+/-2; (d) (-inf,-2)U(2,inf); (e) (-inf,-2]U[2,inf); (f) (-inf,-2)U(2,inf); (g) (-inf,-sqrt5]U[sqrt5,inf)

Step 1: Solve Parts (a)-(c)

(a) \(x^2-4=0 \Rightarrow x=\pm 2\)
(b) \(-x^2+4=0 \Rightarrow x=\pm 2\)
(c) \(x^2-4=-x^2+4 \Rightarrow x=\pm 2\)

Step 2: Solve Parts (d)-(g)

(d) \(x^2-4>0\): \(x<-2\) or \(x>2\)
(e) \(-x^2+4\leq 0\): \(x\leq -2\) or \(x\geq 2\)
(f) \(x^2-4>-x^2+4\): \(x<-2\) or \(x>2\)
(g) \(x^2-4\geq 1\): \(x\leq -\sqrt{5}\) or \(x\geq \sqrt{5}\)