Q. 31 - Precalculus Enhanced with Graphing Utilities

Question

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

(a) Solve $f\left(x\right)=0$. 

(b) Solve $g\left(x\right)=0$. 

(c) Solve $f\left(x\right)=g\left(x\right)$. 

(d) Solve $f\left(x\right)>0$.

(e) Solve $g\left(x\right)\le 0$. 

(f) Solve $f\left(x\right)>g\left(x\right)$. 

(g) Solve $f\left(x\right)\ge 1$.

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

Step-by-Step Solution

Answer

(a) Solution of $f\left(x\right)=0$ is $x=-1 \& 2$

(b) Solution of $g\left(x\right)=0$ is $x=-2 \& 1$

(c) Solution of $f\left(x\right)=g\left(x\right)$ is$x=0$

(d) Solution of $f\left(x\right)>0$ is $\left(-\infty ,-1\right)\cup \left(2,\infty \right)$

(e) Solution of $g\left(x\right)\le 0$ is $\left[-2,1\right]$

(f) Solution of $f\left(x\right)>g\left(x\right)$ is $(-\infty ,0)$

(g) Solution of $f\left(x\right)\ge 1$is $(-\infty ,-1.3]\cup [2.3,\infty )$

Step 1. Given information

The given functions are 

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

Step 2. Part (a)

Substitute zero for function value in function $f\left(x\right)$

$$\begin{gathered} f\left(x\right)=x^2-x-2 \\ 0=x^2-x-2 \\ 0=(x+1)(x-2) \\ x=-1 \& 2 \end{gathered}$$

So $f\left(x\right)=0$ at $x=-1 \& 2$

Step 3. Part (b)

Substitute zero for function value in function $g\left(x\right)$

$$\begin{gathered} g\left(x\right)=x^2+x-2 \\ 0=x^2+x-2 \\ 0=(x-1)(x+2) \\ x=1 \& -2 \end{gathered}$$

So $g\left(x\right)=0$ at $x=-2 \& 1$

Step 4. Part (c)

Equate both functions

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ {x}^2-x-2=x^2+x-2 \\ -x=x \\ x=0 \end{gathered}$$

So $f\left(x\right)=g\left(x\right)$ at $x=0$

Step 5. Pard (d)

Plot the graph of $f\left(x\right)=x^2-x-2$

The graph state that the $f\left(x\right)>0$ at $\left(-\infty ,-1\right)\cup \left(2,\infty \right)$

Step 6. Part (e)

Plot the graph of $g\left(x\right)=x^2+x-2$

The graph state that the solution of $g\left(x\right)\le 0$ is $\left[-2,1\right]$

Step 7. Part (f)

Plot the graph of $f\left(x\right) \& g\left(x\right)$

The graph state that the solution of $f\left(x\right)>g\left(x\right)$ is $\left(-\infty ,0\right)$

Step 8. Part (g)

Plot the graph of $f\left(x\right)$and $y=1$

The graph state that the solution of $f\left(x\right)\ge 1$ is $(-\infty ,-1.3]\cup [2.3,\infty )$