Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

$\left\{x|-1<x<1\right\}$In Problems 25–32, use the given functions f and g. 

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

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

$$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$

Step-by-Step Solution

Answer

(a) $x=-1,1$

(b) $x=-\frac{1}{4}$

(c) $x=-4,0$

(d) $\left\{x|-1<x<1\right\}$

(e) $x\le -\frac{1}{4}$

(f) $\left\{x|-4<x<0\right\}$

(g) $x=0$

Part (a) Step 1. Given information

• The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.
• The equation to solve is $f\left(x\right)=0$

Part (a) Step 2: Plot the graph and observe

• Plot the graph of the function f.

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

Part (b) Step 1. Given information

• The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.
• The equation to solve is $g\left(x\right)=0$

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

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

• From the graph, it can be observed that  $g\left(x\right)=0$ when $x=-\frac{1}{4}$
  

Part (c) Step 1. Given information

The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.

The equation 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 $(-4,-15),(0,1)$.
• So, the given equation holds when $x=-4,0$

Part (d) Step 1. Given information

The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.

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

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

• The inequality holds 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 $-1<x<1$.
• So, the solution set is $\left\{x| -1<x<1\right\}$.

Part (e) Step 1. Given information

The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.

The inequality to solve is  $g\left(x\right)\le 0$.

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

• The inequality holds when the line of the function is on or below the horizontal axis.
• From the graph, the line is on or below the axis for $x\le \frac{1}{4}$.

Part (f) Step 1. Given information

• The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$
• The inequality to solve is $f\left(x\right)>g\left(x\right)$.

Part (f) Step 2. Read the graph

• The inequality holds 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 $-4<x<0$.
• So, the solution set of the inequality is $\left\{x|-4<x<0\right\}$.

Part (g) Step 1. Given information

• The given functions are $$\begin{gathered} f\left(x\right)=-x^2+1 \\ g\left(x\right)=4x+1 \end{gathered}$$.
• The inequality to solve is $f\left(x\right)\ge 1$.

Part (g) Step 2. Read the graph

• The inequality holds when the curve lies on or above the value 1 on the vertical axis.
• From the graph, the curve is above 1 when $x=0$.