Q. 36 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

Part (a) $x=7$.

Part (b) $\left\{x|-4<x\le 7\right\}$.

Part (a) Step 1. Given information.

The given figure is as follows,

There are three lines $f\left(x\right), g\left(x\right), h\left(x\right)$.

Part (a) Step 2. Explanation.

The equation $f\left(x\right)=g\left(x\right)$ represent the intersection of graph $f\left(x\right)$ and $g\left(x\right)$.

Here  intersection of graph $f\left(x\right), g\left(x\right)$ is $\left(7,-8\right)$, the value of x is 7.

Hence, solution of  $f\left(x\right)=g\left(x\right)$ is $x=7$.

Part (b) Step 1. Explanation.

The points $\left(-4,7\right), \left(0,7\right)$  are on $h\left(x\right)$.

The points $\left(0,-8\right), \left(7,-8\right)$ are on $g\left(x\right)$.

The points $\left(-4,7\right), \left(7,-8\right)$ are on $f\left(x\right)$.

The values of $h\left(x\right)$ are greater than values of  function g(x). 

The function f(x) is intersecting h(x) at $\left(-4,7\right)$  and $g\left(x\right)$ at $\left(7,-8\right)$.

For $g\left(x\right)<f\left(x\right)\le h\left(x\right)$ , here lower value of x is -4 and upper value is 7.

So solution of inequality is $\left\{x|-4<x\le 7\right\}$.