# Relations and Functions

INSTRUCTOR GUIDANCE EXAMPLE: Week Five Discussion

Relations and Functions

pg 709 #31

The relation is f(x) = √(x – 1) where the -1 is included with x under the radical.

f(1) = √(1 – 1) = √0 = 0 f(5) = √(5 – 1) = √4 = 2

f(5/4) = √(5/4 – 1) = √(¼) = ½ f(10) = √(10 – 1) = √9 = 3

f(2) = √(2 – 1) = √1 = 1

x f(x) = √(x – 1)

1 0

5/4 ½

2 1

5 2

10 3

This being the graph of a square root function, it is shaped like half of a parabola laid

over on its side. It has a starting point at (1, 0) and extends infinitely in the positive

infinity direction (to the right). This graph falls entirely in the first quadrant and doesn’t

exist in any of the other three quadrants.

This function has no y intercepts and the only x intercept is the same as its starting point

at (1, 0). I chose my x values carefully so that the y values were rational numbers coming

out of the radical. The domain is all real numbers greater than or equal to 1, while the

range is all non-negative real numbers. In interval notation it is written as follows:

D = [1, ) and R = [0, )

This relation is a function because every element of the domain has one and only one

value associated with it in the range, and it passes the vertical line test.

pg711 #57

The relation is f(x) = 1 – |x| where the 1 and the subtraction are outside of the absolute

value bars.

f(-4) = 1 – |-4| = 1 – 4 = -3 f(1) = 1 – |1| = 1 – 1 = 0

f(-1) = 1 – |-1| = 1 – 1 = 0 f(4) = 1 – |4| = 1 – 4 = -3

f(0) = 1 – |0| = 1 – 0 = 1

x f(x) = 1 – |x|

-4 -3

-1 0

0 1

1 0

4 -3

This is the graph of an absolute value function so it is shaped like a straight sided V with

a sharp turn at the vertex. The a value is negative and so the V opens downward. The

majority of the graph occurs in the third and fourth quadrants with only a tiny bit crossing

into the first and second quadrants.

The highest point on the graph occurs at the vertex, (0, 1) and the graph is symmetrical

across the y-axis. The y-intercept is at the vertex (0, 1) as well, and the x intercepts are at

(-1, 0) and (1, 0). The domain is all real numbers while the range is restricted to real

numbers less than or equal to 1.

This relation is also a function because it passes the vertical line test, and each domain

value has one and only one range value associated with it.

Let us take this same function and shift it three units upwards and 4 units to the left.

How will this affect the equation?

Three units upwards means we add a positive 3 outside of the absolute value bars.

Four units to the left means we add a positive 4 inside of the absolute value bars.

The function will now look like this: f(x) = 1 + 3 – |x + 4| so f(x) = 4 – |x + 4| and this reflects a

transformation of the function three units up and four units to the left.

Just for the sake of a thorough example I am also going to show the other function

transformed in the same manner.

Three units upwards means we add a positive 3 outside of the radical.

Four units to the left means we add a positive 4 inside of the radical.

The function will now look like this: f(x) = 3 + √(x – 1 + 4) so f(x) = 3 + √(x + 3)

This does incorporate the two shifts into the transformed function.

In this discussion, you will be assigned two equations where you will complete a variety of math work related to mathematical functions. Read the following instructions in order and view the example (available for download in your online classroom) to complete this discussion. Please complete the following problems according to your assigned number. (Instructors will assign each student their number.) If your assigned number is

Solve the following Problems

37 2 on pages 708 – 711

There are many ways to go about solving math problems. To solve this problem, you will be required to do some work that will not be included in the discussion point. • First, graph your functions so that you can clearly describe the

graphs in your post. Your graph itself is not required in your post, although a discussion of the graph is required. Make sure you have at least five points for each equation to graph. Show all math work for finding the points.

• Mention any key points on the graphs, including intercepts, vertex, or start/end points. Points with decimal values need not be listed, as they might be found in a square root function. Stick to integer value points.

• Discuss the general shape and location of each of your graphs. • State the domain and range for each of your equations. Write

them in interval notation. • State whether each of the equations is a function or not giving

your reasons for the answer.

• Select one of your graphs and assume it has been shifted three units upward and four units to the left. Discuss how this transformation affects the equation by rewriting the equation to incorporate those numbers.

• Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing the thought behind your math work.

◦ Function ◦ Relation ◦ Vertical Line test ◦ Transformation Your initial post should be at least 250 words in length. Support your claims with examples from required material(s) and/or other scholarly resources, and properly cite any references. Respond to at least two of your classmates’ posts by Day 7.