Exercises OAF.0

Click on this link for a file that has Mapping Diagram blanks (2 and 3 axes) for use in the following exercises.
You may want to try this using a spreadsheet. You can download a spreadsheet template for some rational functions by clicking here.
Download worksheet for this section: Worksheet.OAF.pdf.

For each positive power function complete the function table and then create the corresponding mapping diagram and locate the "extreme point" if one exists.

$x$	$f(x)=2x^3$
2
1
0
-1
-2

$x$	$f(x)= -2x^3$
2
1
0
-1
-2

$x$	$f(x)= -(x+1)^4 + 1$
2
1
0
-1
-2

$x$	$f(x)= (x-1)^4-1$
2
1
0
-1
-2

For each negative power function complete the function table and then create the corresponding mapping diagram and locate the "pole" .

$x$	$f(x)= x^{-2}$
2
1
0
-1
-2

$x$	$f(x)=-x^{-3}$
2
1
0
-1
-2

$x$	$f(x)= (x+1)^{-2} -1 $
2
1
0
-1
-2

$x$	$f(x)=(x-1)^{-3}+1$
2
1
0
-1
-2

For the functions in problem 1 c and 1 d, find any values of $a$ where $f(a) = 0$.
For the functions in problem 2 c and 2 d, find any values of $a$ where $f(a) = 0$.
Suppose $f$ is a polynomial function of degree 4 with $f(x)=A(x-2)^2 (x-1)(x+1)$ and $f(0) = 6.
      a. Determine $A$ and the roots of $f$.
      b. Find $f(-2)$ and $f(3)$.
      c. Give the standard polynomial function form for $f$.
      d. Exhibit the results of your work on a graph and a mapping diagram.
Suppose $f$ is a linear fractional rational function with $f(x)= \frac {2x-a} {x-b}$ with its pole at $x=1$ with $f(0)= 3$.
      a. Determine $a$ and $b$.
      b. Use the values of $a$ and $b$ to find $f(-1)$ and $f(2)$.
      c. Give the polynomial - proper rational function form for $f$ based on the given information.
      d. Solve the equation $f(x) = 0$ and display the solutions on your mapping diagram.
For each of the following cubic polynomial functions, create a mapping diagram for the function treated as a composition of core linear functions and the core power function $P_3(x)=x^3$.
a. $f(x)=2x^3 + 1$ b. $f(x)= -2(x-1)^3 - 3$
c. $f(x)=\frac 1 2 x^3 + \frac 1 2$ d. $f(x)=-\frac 1 2 (x+1)^3 + \frac 1 2$
For each of the following rational functions create a mapping diagram for the function treated as a composition of core linear functions and the core power function $P_3^{-1}(x)=\frac 1 {x^3} = x^{-3}$.
a. $f(x)=2x^{-3} + 1$ b. $f(x)= -2(x-1)^{-3} - 3$
c. $f(x)=\frac 1 {2 x^3}+ \frac 1 2$ d. $f(x)=-\frac 1 {2 (x+1)^3} + \frac 1 2$
For each of the following functions use "socks and shoes" to find and create a mapping diagram for its inverse function for the domain $(-\infty,\infty)$.
a. $f(x)=2x^3 + 1$ b. $f(x)= -2(x-1)^3 - 3$
c. $f(x)=\frac 1 {2 x^3}+ \frac 1 2$ d. $f(x)=-\frac 1 {2 (x+1)^3} + \frac 1 2$