Chapter 3.1, Problem 35AYU

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: $g\left(x\right)\le f\left(x\right)<h\left(x\right)$.

To determine

To solve: Solve the following function by using the given graph:

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

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

Answer to Problem 35AYU

Solution:

a. $x=-6$

b. $-6\le x<5$

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 $(-6,-5)$ is the equilibrium point.

Therefore, at $(-6,-5)$, we have $f(x)=g(x)$.

Thus, on solving the given equation, we get the solution as $x=-6$.

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

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

We know that when $f(x)=g(x)$ we have $x=-6$.

Therefore, when $f(x)\ge g(x)$, we get $x\ge -6$.

This can also be written as $-6\le x$ ----- (1)

The equilibrium point for $f(x)$ and $h(x)$ is $(5,12)$.

Therefore, when $f(x)=h(x)$ we get $x=5$.

Thus, when $f(x)<h(x)$, we have $x<5$ ----- (2)

Now, on combining equations (1) and (2), we get the solution for $g(x)\le f(x)<h(x)$ as $-6\le x<5$.