# Inequalities of combined functions

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##### Intros
###### Lessons
1. Introduction to inequalities of combined functions

i. What are inequalities of combined functions?

ii. How many ways can it be solved?

##### Examples
###### Lessons
1. Evaluating Inequalities of Combined Functions by Comparing the Functions Graphically

Let $f(x) = 2x^{2}$ and $g(x) = 3x + 2$.

1. Graph the functions on the same set of axes. Identify the points of intersection.
2. Illustrate the regions for which

i. $f(x)$ > $g(x)$

ii. $g(x)$ > $f(x)$

2. Evaluating Inequalities of Combined Functions by Analyzing the Difference Function

Let $f(x) = 2x^{2} + x - 3$ and $g(x) = x^{2} + x + 13$.

1. Graph the difference function.
2. Illustrate the regions for which

i. $f(x)$ > $g(x)$

ii. $g(x)$ > $f(x)$

3. Let $f(x) = (x-3)(x+5)$ and $g(x) = (x+1)(x-4)$
1. Graph the difference function.
2. Illustrate the regions for which

i. $f(x)$ > $g(x)$

ii. $g(x)$ > $f(x)$

4. Evaluating Inequalities of Combined Functions by Analyzing the Quotient Function

Let $f(x) = (x+3)^{6}$ and $g(x) = (x+3)^{4}$

1. Graph the quotient function.
2. Illustrate the regions for which

i. $f(x)$ > $g(x)$

ii. $g(x)$ > $f(x)$

5. Application of Inequalities of Combined Functions

Nick is starting his own phone company. The cost of making and storing phones can be modelled by the function:

$C(n) = 1.2n + \frac{150,000}{n}$

where $n$ = number of phones. The storage capacity of the company's warehouse is 500 units.

1. Use graphing technology to graph C($n$). What is the domain of this function?
2. Determine the number of phones that can be made if Nick wants to keep the cost below \$1000.
###### Topic Notes

Difference function:

$f(x)$ > $g(x)$$f(x) - g (x)$ > $0$

Quotient function:

$f(x)$ > $g(x)$$\frac{f(x)}{g(x)}$ > $1$