4. Function Representations - Amazon S3 · 4.3 Tabular Representations • Another way to represent...
Transcript of 4. Function Representations - Amazon S3 · 4.3 Tabular Representations • Another way to represent...
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4. Function Representations
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4.1 Plotting Functions
4.2 Return to Function Algebra
4.3 Tabular Representations
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4.1 Plotting Functions
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4.1.1 Plotting Functions with Symmetry and Transformations
4.1.2 Plotting Functions with Asymptotes and Extrema
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4.1 Plotting Functions with Symmetry and Transformations
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• 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))
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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
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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
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Identify as even, odd, or neither:
f(x) =x
4
f(x) =e
x
f(x) =
3px
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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
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Plot f(x) = �(x+ 2)
2
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Plot f(x) = ln(1� x)
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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)
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4.1.2 Plotting Functions with Asymptotes and Extrema
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• 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.
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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.
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Identify asymptotes of f(x) =
2x+ 1
x� 3
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Extrema• Minimums are points
where a function achieves its smallest output.
• Maximums are points where a function achieves its largest output.
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Find the extrema of f(x) = sin(x) + 2,
0 x 2⇡
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4.2 Return to Function Algebra
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• 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)
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Composition of Functions
(f � g)(x) = f(g(x))
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For each of the following function pairs, compute (f � g)(x), (g � f)(x) :
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f(x) = sin(x)
g(x) =x
2
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f(x) =e
x
g(x) =x
3
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f(x) = log2(x)
g(x) =|x|
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4.3 Tabular Representations
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• 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))
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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
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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