3.4 Chapter Summary

This chapter introduced a toolkit for visualizing functions.

Interactive Tools:¶

1. Function Properties and Linear Transformations - Use this tool to visualize linear transformations of functions (inputs and outputs) and to check your understanding of key global properties.

2. Function Combination - Use this tool to visualize linear combinations and products of functions.

3. Function Composition I - 2D - Use this tool to visualize function compositions.

4. Function Composition II - 3D - Use this tool to visualize function compositions in three dimensions. It works best when the outer function is monotonic and bounded.

5. Function Inversion - Use this tool to visualize function inverses.

Global Function Properties¶

Definitions and examples are all available in Section 3.1.

1. Continuity: A function $f$ is continuous if $\lim_{x \rightarrow x_*} f(x)$ is well defined for all $x_*$ and equals $f(x_*)$.

2. Differentiable: A function $f$ is differentiable if $\frac{d}{dx} f(x)$ exists for all $x$. That is, if it has a well-defined slope everywhere.

3. Symmetric: A function is even if $f(-x) = f(x)$. A function is odd if $f(-x) = -f(x)$.

   • A function is symmetric about some axis of symmetry if there exists an $x_*$ such that $f(-(x - x_*)) = f(x - x_*)$ or $f(-(x - x_*)) = - f(x - x_*)$. That is, if $f$ is even or odd with respect to reflections about $x_*$.

4. Bounded: A function is bounded from below if $f(x) \geq m$ for some $m$. A function is bounded from above if $f(x) \leq M$ for some $M$.

5. Monotonic: A function $f$ is monotonic if it never changes direction.

6. Convex/Concave: A function is convex if it ``bends upwwards." It is concave if it “bends downwards.”

Function Operations¶

Definitions and examples are all available in Section 3.2.

1. Linear Transformations...

   1. to the Input:

      • Horizontal Translation: Replace $f(x)$ with $f(x - s)$ to translate the function horizontally by a shift $s$.

      • Dilation: Replace $f(x)$ with $f(x/a)$ for some $a > 0$ to dilate the function.

   2. to the Output:

      • Vertical Translation: Replace $f(x)$ with $f(x) + h$ to translate the function vertically by height $h$.

      • Vertical Scaling: Replace $f(x)$ with $c f(x)$ to scale the function.

2. Function Combinations:

   1. Algebraic Combination:

      • Function Addition and Multiplication: As they sound. $f(x) + g(x)$ or $f(x) \times g(x)$.

      • Linear Combination: This is a special version of function addition. It looks like $a f(x) + b g(x)$ for some coefficient $a$ and $b$ that scale each term in the combination.

   2. Function Composition: The composition of $f$ and $g$ is $f \circ g(x) = f(g(x))$. Many distributions are expressed as compositions.

3. Inverses: If $f$ is monotonic, then its inverse, $f^{-1}$ is the function that accepts outputs of $f$ and returns the matching input.

   • In other words, given $f(x) = y$, $f^{-1}(y) = x$.

Local Function Properties:¶

Definitions and examples are all available in Section 3.3.

If you can’t draw a function using its global properties, or by recognizing it as a transformation/combination of functions you recognize, then you will need to evaluate it at a series of key reference points.

1. Make a list of reference points. In rough order of work to evaluate.

   • Any inputs where $f$ is easy to evaluate (often, $x = 0$ and $x = \pm 1$).

   • The smallest and largest possible inputs. If necessary, take limits.

   • An axis of symmetry if it exists.

   • Any roots (locations where $f(x) = 0$) or poles (locations where $f(x)$ diverges).

   • Any points of discontinuity or nondifferentiability.

   • All maxima and minima of the function.

     • Check for places where the derivative $\frac{d}{dx} f(x) = 0$ and the sign of the derivative in between its roots

   • Any inflection points of the function.

     • Check for places where the second derivative $\frac{d^2}{dx^2} f(x) = 0$ and the sign of the second derivative in between its roots

2. Then:

   • Evaluate $f$ at each reference.

   • Evaluate the slope of $f$ at each reference. Check whether you know the slope before computing a derivative.

   • Evaluate the sign of the second derivative of $f$ at each reference. Check whether you know the sign of the second derivative before computing a derivative.

That’s a long list, but you rarely need all of it. The more you practice, the faster you’ll get at recognizing functions, and the fewer reference points you’ll need. Most functions don’t have all the references on this list, so you can usually skip a bunch with little effort. For many distributions it is enough to check the largest and smallest possible inputs, check for symmetry, and to identify maxima and minima.

On Plotting Standards¶

This chapter has established three tools for plotting: (1) check global behavior, (2) check composition, (3) check local behavior at a list of references. Each works by identifying some summary characteristic of a function that directs its behavior. This list is pretty exhaustive. If you do (1), (2), and (3), you’ll be able to draw an accurate picture.

Make sure that, when you solve for and draw the summary features that are numbers (e.g. values of $f$ at references, or position of references), you should solve for those features leaving all free parameters of your function free, and, when you plot them, plot them precisely. Plotting precisely doesn’t mean getting every value of $f$ in exactly the right location on your graph. It means getting the key reference points right.

We will hold you to high plotting standards in this class. Take time on your HW this week to draw carefully. Work methodically. Buy some graph paper, or use a graph paper background on your tablet if you have a tablet. Imagine you are an artist, architect, designer, engineer, or draftsman. We are looking for quality work. Going slowly through the five functions on HW 3 is great practice. Do these slowly so you can work efficiently in the future without sacrificing quality.

Drawing functions is more than a physical task or simple practice. If you always rush your illustrations it will be hard to hold a function in your mind’s eye when you read a formula. One of our true aims in this class is to help you recognize the shapes encoded by formula. Strings of letters, expressed in Greek, are not very memorable. Images are extremely memorable. More importantly, the shape of a function is its soul, not the symbols that express it. If you want to model some data, you will need to pick a formula, so you will need to be able to sift through an album of mental images, each tagged to some formula. Alternately, if you solve a problem and get out a function that answers your question, you should be able to close your eyes and see the answer, not as a string of symbols, or a single number, but as a curve that expresses all of the different possible answers to your problem as functions of the parameters of the problem.

We are asking you to do the physical task now since working slowly is a memorable experience. Physically drawing a function yourself, instead of leaving it to a computer, forces you to actually embody the mechanics of the function. That motor memory should stick. Repeat it slowly a couple times, and, just like hitting a tennis ball, or playing an instrument, your brain will adapt. Soon you’ll be able to visualize the process without acting it out. Then you’ve acquired the desired skill.