# Mat 222 week 5 written assignment

Read the following instructions in order to complete this assignment:

1. We define the following functions:

f(x)=2x+5 g(x)=x^2-3 h(x)=7-x

3

Compute (*f *– *h*)(4).

o Evaluate the following two compositions: (fog)(x) and (hog)(x) .

o Transform the *g(x)* function so that the graph is moved 6 units to the right and 7 units down.

o Find the inverse functions f^{-1} (x) and ^{h-1}(x).

2. Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the example and be concise in your reasoning. In the body of your essay, please make sure to include:

- Your solutions to the above problems, making sure to include all mathematical work for both problems, as well as explaining each step.
- A discussion of the applicability of functions to the real world, based upon your reading of Chapter 11 of
*Elementary and Intermediate Algebra*. Be sure to use specific examples, a brief discussion of why your examples are important, and to cite your sources.

INSTRUCTOR GUIDANCE EXAMPLE: MAT 222 Week 5 Written Assignment

Composition and Inverse

We are working with the following functions:

** f(x)** = 5

**– 3**

*x***=**

*g(x)*

*x*^{2}+ 2

**) = 3 +**

*h(x*

*x*7

We have been asked to compute (** f – h**)(4).

(** f – h**)(4) =

**(4) –**

*f***(4) So we can evaluation each separately and then subtract.**

*h*** f**(4) = 5(4) – 3

= 20 – 3 = 17 ** f**(4) = 17

** h**(4) = (3 + 4)/7

= 7/7 = 1 ** h**(4) = 1

(** f – h**)(4) = 17 – 1 = 16 This is our answer.

Next we are to compose two pairs of the functions into each other. First we will work out

(** f **°

**)(**

*g***) =**

*x***(**

*f***(**

*g***)) This means the rule of**

*x***will work on**

*f***.**

*g* = ** f**(

*x*^{2}+ 2) Here

**is now going to work on the rule of**

*f***.**

*g* = 5(*x*^{2} + 2) – 3 The rule of ** f** is applied to

**g**.

= 5*x*^{2} + 10 – 3 Simplifying.

(** f **°

**)(**

*g***) = 5**

*x*

*x*^{2}+ 7 The final results

Now we will compose the following:

(** h **°

**)(**

*g***) =**

*x***(**

*h***(**

*g***)) The rule of**

*x***will work on**

*h***.**

*g* = ** h**(

*x*^{2}– 3)

= 3 + (*x*^{2} + 2) The rule of ** h** is applied to

**.**

*g*7

(** h **°

**)(**

*g***) = 5 + x**

*x*^{2}The final results.

7

Next we are asked to transform ** g(x)** so that the graph is placed 6 units to the right and 7 units downward from where it would be right now.

Six units to the right means to put -6 in with x to be squared.

Seven units downward means to put -7 outside of the squaring.

The new function will look like this:

** G**(

**) = (**

*x***– 6)**

*x*^{2}+ 2 – 7

** G**(

**) = (**

*x***– 6)**

*x*^{2}– 5

Our last job is to find the inverse of two of our functions, ** f** and

**. To find the inverse we will write the function with y instead of the function name, then we will switch the places of x and y, and solve for y again. Here are the functions:**

*h* ** f(x)** = 5

**– 3**

*x***) = 3 +**

*h(x*

*x*7

Here we replace ** f(x**) and

**) with**

*h(x*

*y*:** y** = 5

**– 3**

*x***= 3 +**

*y*

*x*7

Here we switch the y and the x:

** x** = 5

**– 3**

*y***= 3 +**

*x*

*y*7

Now we solve for ** y**:

Add 3 to both sides. Multiply both sides by 7.

** x** + 3 = 5

**7**

*y***= 3 +**

*x*

*y*One more solving step:

Divide both sides by 5. Subtract 3 from both sides..

** x** + 3 =

**7**

*y***– 3 =**

*x*

*y*5

** y** =

**+ 3**

*x***= 7**

*y***– 3**

*x*5

Presenting the inverse functions:

f ^{-1}(x) = x + 3 h^{-1}(x) = 7x – 3

5

Thus we have evaluated, combined, composed, and found inverses of our functions.

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