Chapter 3.1, Problem 36AYU

In parts (a) and (b), use the following figure.

a. Solve the equation: $f\left(x\right)=g\left(x\right)$.

b. Solve the inequality: $g\left(x\right)<f\left(x\right)\le h\left(x\right)$.

To determine

To calculate: Solve the following equation by using the given graph:

a. $f(x)=g(x)$

b. $g(x)<f(x)\le h(x)$

Answer to Problem 36AYU

Solution:

a. $x=7$

b. $7<x\le -4$

Explanation of Solution

Given:

The given figure is

Formula Used:

The points in a graph are written in the form $\left( x,f(x) \right)$

The equilibrium point of 2 functions is the point at which the functions are equal and therefore, the 2 functions meet at the equilibrium point.

Calculation:

a. We have to solve the equation $f(x)=g(x)$.

From the graph, we can see that $(7,-8)$ is the equilibrium point.

Therefore, at $(7,-8)$, we have $f(x)=g(x)$.

Thus, on solving the given equation, we get the solution as

$x=7$

b. We have to solve the equation $g(x)<f(x)\le h(x)$ .

On splitting the above inequalities, we get $f(x)>g(x)$and $f(x)\le h(x)$.

We know that when $f(x)=g(x)$ we have $x=7$.

Therefore, when $f(x)>g(x)$, we get $x>7$.

This can also be written as $7<x$ ----- (1)

The equilibrium point for $f(x)$ and $h(x)$ is$(-4,7)$.

Therefore, when $7<x\le -4$, we get $x=-4$.

Thus, when $f(x)\le h(x)$, we have $x\le -4$ ----- (2)

Now, on combining equations (1) and (2), we get the solution for $g(x)\le f(x)<h(x)$ as $7<x\le -4$.