Question

Suppose that $f\left(x\right)=4x-1$and $g\left(x\right)=-2x+5$

(a) Solve $f\left(x\right)=0$. (b) Solve $f\left(x\right)>0$

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

(e) Graph$y=f\left(x\right)$and$y=g\left(x\right)$and label the point that represents the solution to the equation$f\left(x\right)=g\left(x\right)$.

Short answer

Part (a) $x=\frac{1}{4}$

Part (b) $\left\{x|x>\frac{1}{4}\right\}$

Part (c) $x=1$

Part (d) $\left\{x|x\le 1\right\}$

Part (e)

Step-by-step solution

Part (a). Step 1. Given information.

There are two functions$f\left(x\right)=4x-1$and$g\left(x\right)=-2x+5$.

Part (a). Step 2. Explanation.

To solve $f\left(x\right)=0$equate the function with 0.

$$\begin{gathered} f\left(x\right)=0 \\ 4x-1=0 \\ 4x=1 \\ x=\frac{1}{4} \end{gathered}$$

At$x=\frac{1}{4}$,$f\left(x\right)=0$.

Part (b). Step 1. Explanation.

To solve $f\left(x\right)>0$, We have $f\left(x\right)=4x-1$.

so,

$$\begin{gathered} f\left(x\right)>0 \\ 4x-1>0 \\ 4x>1 \\ x>\frac{1}{4} \end{gathered}$$

For all$x>\frac{1}{4}$,$f\left(x\right)>0$

Part (c). Step 1. Explanation.

To solve $f\left(x\right)=g\left(x\right)$

So,

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ 4x-1=-2x+5 \end{gathered}$$

Combine like terms.

$$\begin{gathered} 6x=6 \\ x=1 \end{gathered}$$

For $x=1$,$f\left(x\right)=g\left(x\right)$

Part (d). Step 1. Explanation.

To solve $f\left(x\right)\le g\left(x\right)$.

So,

$$\begin{gathered} f\left(x\right)\le g\left(x\right) \\ 4x-1\le -2x+5 \\ 6x\le 6 \\ x\le 1 \end{gathered}$$

For all$x\le 1$,$f\left(x\right)\le g\left(x\right)$

Part (e). Step 1. Explanation.

Make table for $y=f\left(x\right)$

x	$y=4x-1$
-2	-9
0	-1
2	7

Make table for $y=g\left(x\right)$

x	$y=-2x+5$
-2	9
0	5
2	1

Graph $y=f\left(x\right)$and$y=g\left(x\right)$and label the point$f\left(x\right)=g\left(x\right)$

$f\left(x\right)=g\left(x\right)$is intersection of graph$f\left(x\right)$and$g\left(x\right)$which is$\left(1,3\right)$