Chapter 3.1, Problem 29AYU

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

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

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

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

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

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

To determine

Solve the following the function:

a. Solve $f(x)=0$

b. Solve $f(x)>0$

c. Solve $f(x)=g(x)$

d. Solve $f(x)\le g(x)$

e. Graph $y=f(x)$ and $y=g(x)$ label the points that represents the solution to the equation $f(x)=g(x)$

Answer to Problem 29AYU

Solution:

a. $x=\frac{1}{4}$

b. $x>\frac{1}{4}$

c. $x=1$

d. $x\le 1$

e. The graph is as shown below:

Explanation of Solution

Given:

It is given that $f(x)=4x-1$ and $g(x)=-2x+5$.

a. We need to solve for $f(x)=0$.

$f(x)=0$

$\Rightarrow 4x-1=0$

$\Rightarrow 4x=1$

$\Rightarrow x=\frac{1}{4}$

b. We need to solve for $f(x)>0$.

$f(x)>0$

$\Rightarrow 4x-1>0$

$\Rightarrow 4x>1$

$\Rightarrow x>\frac{1}{4}$

c. We need to solve for $f(x)=g(x)$.

$f(x)=g(x)$

$\Rightarrow 4x-1=-2x+5$

$\Rightarrow 4x+2x=5+1$

$\Rightarrow 6x=6$

$\Rightarrow x=1$

d. We need to solve for $f(x)\le g(x)$.

$f(x)\le g(x)$

$\Rightarrow 4x-1\le -2x+5$

$\Rightarrow 4x+2x\le 5+1$

$\Rightarrow 6x\le 6$

$\Rightarrow x\le 1$

e. Now, we have to graph $y=f(x)$ and $y=g(x)$.

$(1,3)$ is the point that represents the solution to the equation $f(x)=g(x)$.