Chapter 3.3, Problem 65AYU

In Problems 77-82, for the given functions $f\text{ and }g$,

(a) Graph $f\text{ and }g$ on the same Cartesian plane.

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

(c) Use the result of part (b) to label the points of intersection of the graphs of $f\text{ and }g$.

Shade the region for which $f\left(x\right)>g\left(x\right)$, that is, the region below $f$ and above $g$.

$f\left(x\right)=-x^2+4;g\left(x\right)=-2x+1$

To determine

a. Graph $f$ and $g$ on the same Cartesian plane.

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

c. Label the points of intersection.

d. Shade the region below $f$ and above $g$.

Answer to Problem 65AYU

a.

b. The point of intersection is $(3,-5)$ and $(-1,3)$.

c.

d.

Explanation of Solution

Given:

The functions are

$f(x)=-x^2+4$

$g(x)=-2x+1$

Calculation:

a. The graph of the given functions are

b. Now, we have to solve for $f(x)=g(x)$. Therefore, we have

$-2x+1={\mathrm{- x}}^2+4$

$\Rightarrow x^2-2x-3=0$

$\Rightarrow (x-3)(x+1)=0$

This implies that

$x-3=0$   $x+1=0$

and

$\Rightarrow x=3$   $x=-1$

At $x=3$, we have $g(x)\text{ = }-2(3)+1=-5$

At $x=-1$, we have $g(x)\text{ = }-2(-1)+1=3$.

Thus, we get the point of intersection as $(3,-5)$ and $(-1,3)$.

c. The label the point of intersection is as shown below:

d. The shaded region below $f$ and above $g$ is shown below: