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

$$\begin{gathered} \left(a\right)f\left(x\right)=0 \\ \left(b\right)g\left(x\right)=0 \\ \left(c\right)f\left(x\right)=g\left(x\right) \\ \left(d\right)f\left(x\right)>0 \end{gathered}$$

role="math" localid="1646199855131" $$\begin{gathered} \left(e\right)g\left(x\right)\le 0 \\ \left(f\right)f\left(x\right)>g\left(x\right) \\ \left(g\right)f\left(x\right)\ge 1 \end{gathered}$$

The given function is$f\left(x\right)=x^2-1$.

• Plot the graph of the function.

• From the graph, it can be observed that$f\left(x\right)=0$when$x=-1,1$.

The second given function is$g\left(x\right)=3x+3$.

• Plot the function in the graph obtained for the first function.

• From the graph, it can be observed that $g\left(x\right)=0$when$x=-1$.

The given equality to solve is$f\left(x\right)=g\left(x\right)$.

• For $f\left(x\right)=g\left(x\right)$, the curves of both the functions must intersect.
• From the graph, it can be observed that the functions intersect at $(-1,0)\mathrm{and}(4,15)$.
• So,$f\left(x\right)=g\left(x\right)$at$x=-1,4$.

The given inequality is$f\left(x\right)>0$.

• $f\left(x\right)>0$ when the curve of the function is above the horizontal axis.
• According to the graph obtained in step 2 of part (b), the curve is above the horizontal axis when $x<-1orx>1$.
• So, the solution set is$\left\{x:x<-1orx>1\right\}$.

The given inequality is$g\left(x\right)\le 0$

• $g\left(x\right)\le 0$when the curve of the function is on or below the horizontal axis.
• From the graph, the line is on or below the axis for$x\le -1$.

The given inequality is $f\left(x\right)>g\left(x\right)$.

• The function $f\left(x\right)>g\left(x\right)$when the curve lies above the line on the graph.
• From the graph, it can be observed that the curve is above the line when $x<-1orx>4$.
• So, the solution set of the inequality is$\left\{x:x<-1orx>4\right\}$

The given inequality is $f\left(x\right)\ge 1$.

• The inequality holds when the curve lies above the value 1 on the vertical axis.
• From the graph, the curve is above 1 when $x\le -\sqrt{2}orx\ge \sqrt{2}$.
• So, the solution set of the inequality is$\left\{x:x\le -\sqrt{2}orx\ge \sqrt{2}\right\}$.