Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

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

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

$$\begin{gathered} \left(e\right) g\left(x\right)\le 0 \\ \left(f\right) f\left(x\right)>g\left(x\right) \\ \left(g\right) f\left(x\right)\ge 1 \end{gathered}$$

Step-by-Step Solution

Answer

The required solutions are:

(a) $x=-1,1$

(b) $x=-1$

(c) $x=-1,4$

(d) $\left\{x:x<-1 or x>1\right\}$

(e) $x\le -1$

(f) $\left\{x:x<-1 or x>4\right\}$

(g) $\left\{x:x\le -\sqrt{2}or x\ge \sqrt{2}\right\}$

Part (a) Step 1. Given Information

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

Part (a) Step 2. Plot the function and observe

• Plot the graph of the function.

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

Part (b) Step 1. Given information

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

Part (b) Step 2. Plot the function and observe

• 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 $x=-1$.

Part (c) Step 1. Given Information

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

Part (c) Step 2. Read the Graph

• 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 $x=-1,4$.

Part (d) Step 1. Given Information

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

Part (d) Step 2. Find the region above the horizontal axis.

• $f\left(x\right)>0$ 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 $x<-1 or x>1$.
• So, the solution set is $\left\{x:x<-1 or x>1\right\}$.

Part (e) Step 1. Given Information

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

Part (e) Step 2. Find the region on or below the horizontal axis.

• $g\left(x\right)\le 0$ 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 $x\le -1$.

Part (f) Step 1. Given Information

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

Part (f) Step 2. Observe from the graph

• 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 $x<-1 or x>4$.
• So, the solution set of the inequality is $\left\{x:x<-1 or x>4\right\}$

Part(g) Step 1. Given information

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

Part(g) Step 2. Observe from the graph

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