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Functions on Real Numbers
Until now our functions have basically been just toys. Here’s where we bring out the big guns!
Since this is math, most of the functions we deal with will have numbers (or subsets of the real
numbers) as their set of inputs.
This line here represents our most common set of inputs, the real numbers.
Challenge: Think of the smallest number you can that is still bigger than 0. I bet I can think of a
smaller one.
(Whatever number you choose, I will just divide it by 2!)
Writing y ∶ ℝ → ℝ , y(x) = 𝑥 + 2, describes the function y with domain ℝ, that takes in a
real number 𝑥, adds 2 to it and then produces the result y of 𝑥, written y(𝑥), which is another
real number.
Last chapter we drew pictures with dots and arrows to represent maps. Since ℝ is infinite,
those pictures aren’t going to cut it this time. Instead, we take another approach:
As pictured to the left; we draw our domain, the real
number line ℝ, horizontally. We draw our range, which
is also the real number line ℝ, vertically, so that it cuts
our domain ℝ at 0, at a right angle.
In this picture, we call the horizontal real number line (representing our domain) the x axis, and
the vertical real number line (representing our range) is called the y axis.
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Now for the interesting part. For each 𝑥 value in our domain, for example 𝑥 = 1, we move
across the 𝑥 axis horizontally to 𝑥 = 1, we then move vertically until we are in line with y(𝑥)
(the number that comes out when we put 𝑥 into our function y) on the y axis.
In this case, y(𝑥 ) = 𝑥 + 2 , so the output for 𝑥 = 1 is y(1) = 1 + 2 = 3.
So we go across to 𝑥 = 1 on our 𝑥 axis, then we move vertically until we are in line with
y(1) = 3 on our y axis. We then put a dot there, as pictured:
The blue dot represents the pair (1, 3).
This means that 1 is sent to 3 by the function y.
Repeating this process for a bunch of other 𝑥 values (values in our domain), we find that all the
dots for this function are part of a straight line.
Remember, our domain is the entire real number
line now, so we can put numbers into the
function y that are not whole.
For example, y(2.5) = 2.5 + 2 = 4.5 ,
y(1.77723) = 1.77723 + 2 = 3.77723.
The more points you plot, the more of the
straight line is filled in.
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Eventually, if we keep plotting points, it will just become a solid blue line.
This makes sense, since y (𝑥 ) = 𝑥 + 2 means y(𝑥 + ℎ) = (𝑥 + ℎ) + 2 = 𝑥 + 2 + ℎ.
In words, moving horizontally by any amount ℎ in the domain moves the range value vertically
by the same amount ℎ.
Hence, we would indeed expect the points to form a line when representing our function with
this type of picture.
The baby-blue line was just for show to help us understand the concepts. Usually we just use a
black line, as in the following examples:
Observe, in particular, that functions don’t have to look like straight lines.
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Exercises
1. Plot the functions y(𝑥 ) = 𝑥 − 2 , y(𝑥 ) = 𝑥 2 and y(𝑥 ) = 2𝑥 .
Hint: Choose a bunch of 𝑥 values and put them into the function y , then place dots in
the appropriate places. After you have enough dots to get an idea of what the function
looks like, connect them with an appropriate curve.
2. Plot the function y(𝑥 ) = 𝑥. Is it surprising that it makes a 45 degree angle with the 𝑥
axis?
3. Plot the functions 𝑦1 (𝑥 ) = 𝑥 2 − 4𝑥 + 4 and 𝑦2 (𝑥 ) = (𝑥 − 2)(𝑥 − 2). What do you
notice and why do you notice this?
4. Plot the function y(𝑥 ) = 3.
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5. The following picture shows the graph of the function y(𝑥 ) = − 2 𝑥 2 + 4.
When I draw an orange line parallel to the 𝑥 axis and intersecting the y axis at any value
of y less than 4 (the line y = 2 is shown in the picture), it intersects the curve at two
points. Why is this?
6.
For any fixed numbers a and b, what will our picture of the function y = a𝑥 + b look
like?