4. Function Representations

4.1 Plotting Functions

4.2 Return to Function Algebra

4.3 Tabular Representations

4.1 Plotting Functions

4.1.1 Plotting Functions with Symmetry and Transformations

4.1.2 Plotting Functions with Asymptotes and Extrema

4.1 Plotting Functions with Symmetry and Transformations

• Drawing a function in the Cartesian plane is extremely useful in understand the relationship it defines.

• One can always attempt to plot a function by computing many pairs , and plotting these on the Cartesian plane.

• However, simpler qualitative observations may be more efficient. We will discuss of a few of these notions before moving on to some standard function plots to know.

(x, f(x))

Symmetry of Functions• A function is said to be even/is symmetric

about the y-axis if for all values of • Functions that are even are mirror images of

themselves across the -axis.

f(x)x, f(x) = f(�x).

y

Symmetry of Functions• A function is said to be odd/has symmetry

about the origin if for all values of , .

• Functions that are odd can be reflected over the -axis, then the -axis.

f(x)f(�x) = �f(x)x

x

y

Identify as even, odd, or neither:

f(x) =x

4

f(x) =e

x

f(x) =

3px

Transformations of Functions

It is also convenient to consider some standard transformations for functions, and how they manifest visually:

• shifts the function to the left by if is positive, and to the right by if is negative.

• shifts the function up by if is positive, and down by if is negative.

• reflects the function over the -axis. • reflects the function over the -axis.

f(x) 7! f(x+ a) a aa a

f(x) 7! f(x) + b

f(x) 7! f(�x)

f(x) 7! �f(x) x

y

b bb b

Plot f(x) = �(x+ 2)

2

Plot f(x) = ln(1� x)

Solving Equations with Plotting• Consider the generic

equation

• One can using technology to plot the functions, and then look for their intersections.

f(x) = g(x)

4.1.2 Plotting Functions with Asymptotes and Extrema

• We can also consider other features when plotting functions.

• Two important features are asymptotes and extrema.

• Asymptotes are, for us, either vertical or horizontal. The are due to restrictions on the domain or range of a function, respectively.

• Extrema are maximums or minimums of a function.

Asymptotes• Vertical asymptotes occur

where a function has domain restrictions, typically when there is division by 0.

• Horizontal asymptotes occur when a function approaches, but never reaches, a certain output.

Identify asymptotes of f(x) =

2x+ 1

x� 3

Extrema• Minimums are points

where a function achieves its smallest output.

• Maximums are points where a function achieves its largest output.

Find the extrema of f(x) = sin(x) + 2,

0 x 2⇡

4.2 Return to Function Algebra

• Just as with numbers, we can perform algebraic operations on functions.

• Functions can be added, subtracted and multiplied naturally.

• Functions can also be divided, but one must take care to avoid division by 0, which is not a well-defined mathematical operation.

(f + g)(x) = f(x) + g(x)

(f · g)(x) = f(x) · g(x)✓f

g

◆(x) =

f(x)

g(x)

Composition of Functions

(f � g)(x) = f(g(x))

For each of the following function pairs, compute (f � g)(x), (g � f)(x) :

f(x) = sin(x)

g(x) =x

2

f(x) =e

x

g(x) =x

3

f(x) = log2(x)

g(x) =|x|

4.3 Tabular Representations

• Another way to represent functions is a table relating inputs to outputs.

• This is a very natural way to visualize and understand a function’s behavior, as the input and output pairs are explicit.

• It has an obvious disadvantage, in that one must have a very long table to show many pairs.

• It can also be difficult to discern the overall pattern from the table.

• It is, however, convenient for computing values of composed functions of the form, for example,

(x, f(x))

f � g(x) = f(g(x))

x f(x) g(x) f(g(x))

1 2 -4

2 3 -5

3 -1 2

4 2 1

5 4 0

6 6 -1

x f(x) g(x) (f+g)(x) (fg)(x) f(g(x))

-2 1 -1

-1 1 0

0 0 2

1 3 1

2 5 2

3 4 4

4 2 3