Function Theory

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Function Graphs

Discussion

Below is a graph of f(x) plotted for every integer value of x.

What would the graph look like if the function was graphed for all real x rather than integer values only?

The plot of f(x) at integer x already traces out an “S‐shaped” curve—slowly rising in the middle and steep at the ends. If you were to graph f for every real x, instead of just drawing isolated dots, you would draw a single continuous curve through those points with the following features:

• It is strictly increasing for all x.

• Around x≈−2 up to x≈3 it is relatively flat (small slope).

• As x→−∞ and x→+∞ it becomes steeper, so the tails rise more sharply.

• The result is a smooth “S‐shape” passing through all the blue dots.

In other words, replace the discrete dots by a smooth increasing curve interpolating them—shallow through the centre and steep towards the left and right ends—producing a continuous S‐shaped graph.

Graph of a function

The graph is a visual representation of a function made by placing all the points (x,f(x)) on a coordinate plane. Since most functions have infinitely many possible inputs x, the graph often looks like a line.

Discussion

Previously, we defined functions as relations which had no more than one output for any input.

How can you tell visually if a graphed relation is a function?

A relation that assigns more than one output to the same input will have at least two points sharing the same x‐coordinate but different y‐coordinates. Concretely, suppose for some a the relation contains both

(a,y1)and(a,y2)

with y1=y2. On the Cartesian plane these two points lie one above the other at x=a.

For example, the points

(2,1)and(2,3)

both have x=2 but different y–values, so on the graph you would see two dots on the vertical line x=2. That configuration shows the relation gives two outputs for the single input 2.

Vertical line test

As mentioned earlier, a function is a special type of relation where each x value has only one possible y-value.

Graphically, this means functions pass the vertical line test, which means that no vertical line intersects the graph of a function twice:

Checkpoint

Determine which graph(s) above are functions.

Select the correct option

Discussion

f(x) is graphed above. Find f(4).

To find f(4), we look for the point on the curve whose x-coordinate is 4.

The graph shows a solid dot at (4,3), so the corresponding y-value is 3.

f(4)=3

Finding function values from graph

We can find the value f(a) of a function by looking at the y-value of the graph where x=a:

Exercise

The following graph shows the curve of y=f(x).

Find
1. f(−3)
2. f(0)
It is given that f(k)=−1. Find k.

Select the correct option

Graphing with technology

When the graph of a function is not known, a calculator can be used to plot its curve. Under the hood, the calculator is simply plotting points at regular x intervals, and connecting them with straight lines. For a small enough interval, this approximation is basically unnoticeable.

When graphing on a calculator, many important features of the function(s) graphed can be calculated easily.

Intersections with GDC

Calculators are able to provide precise information on coordinates of an intersection of two functions.

Using a TI-84, 2nd > CALC > INTERSECT,

or using a Casio, SHIFT > G-SOLV > INTSECT

will find an intersection between two functions.

Checkpoint

Find the coordinates of intersection of

y=x2−1andy=x+2

Select the correct option

Exercise

Find the coordinates of the intersection(s) of y=x2+1√5x5+2−x and y=x+1.2. Give your answers to three significant figures.

Select the correct option

x-intercepts

The x-intercepts of a function are the values of x where the curve intersects the x-axis. Since the x-axis has the equation y=0, this means the function is equal to zero. For this reason, x-intercepts are also often called the "zeros" of the function.

Checkpoint

Find the x-intercept(s) of y=x3−3.

Select the correct option

y-intercepts

The y-intercept of a function is the values of y where the curve intersects the y-axis. Since the y-axis has the equation x=0, we can plug 0 into the function definition to find the intercept.

Exercise

Find the y-intercept of

f(x)=√8−2x√2x2+18.

Select the correct option

Checkpoint

Find the y-intercept of the function graphed above
Find the x-intercept(s) of the function graphed above.

Select the correct option

In a previous exercise, we solved for the y-intercept of f(x)=√9−3x√3x2+4 by plugging in 0. We found f(0)=32.

Alternatively, with a TI-84:

Under Y= set Y1=√9−3x√3x2+4 and GRAPH . Then, 2nd > CALC > VALUE will let you easily calculate f(x) for any x, including x=0. This is a good method if you need to find many values of a function or need the y-intercept of a function that you have already graphed.

Calculator: finding value of a function

On a graphing calculator, you can quickly calculate f(a) for any function f(x) and any a in its domain.

On a TI-84, under Y= set Y1=f(x) and GRAPH it. Then, 2nd > CALC > VALUE and input a when prompted X=. This returns f(a).

On a Casio, from the calculate app, press FUNCTION, choose Define f(x) and input your equation. Then, press FUNCTION again and now press f(x) , and type in the value of a and a close parenthesis. This returns f(a).

Checkpoint

It is given that f(x)=x2−5x+6x2−1. Find f(2.5).

Select the correct option

Discussion

f(x)=ex is graphed below. The image to right zooms in on the outlined box on the left.

What do you notice about the value of f as x becomes smaller?

As x decreases (moves left), the values of

f(x)=ex

get smaller and smaller but remain positive. In fact one shows

x→−∞limex=0

so the curve approaches the x–axis without ever touching it. In the zoomed‐in view you can see y=ex falling closer and closer to zero as x goes from −2.25 down to −4, but always staying just above the axis.

Horizontal asymptotes

A horizontal asymptote is a horizontal line that a graph approaches but never actually touches or crosses. It's seen where a function approaches a certain constant value as x gets very large (positive or negative).

Horizontal asymptotes have the equation y=a for some a∈R.

To find a horizontal asymptote with a calculator, plot the curve and use trace to inspect the value of y for large (positive or negative) x.

Exercise

Find the equation of the horizontal asymptote of y=√x+1√3x.

Select the correct option

Vertical asymptotes

A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. It's usually seen where a function "blows up," meaning the function’s values become infinitely large or small as you get closer to certain values of x.

Vertical asymptotes have the equation x=a for some a∈R.

Vertical asymptotes occur for functions of the form

f(x)=h(x)g(x)

when h(x)=0 and g(x)=0, as division by zero is undefined, and division by a number close to zero gives a large number.

Checkpoint

Find the equation of the vertical asymptote of y=x3−x+11 using a calculator.

Select the correct option

On the graph above, the point A is special. It is called a local minimum. Even though A is not the lowest point on the graph, it is lower than all of the points surrounding it. One way to think of this is that we could zoom in on the graph above enough so that it would look like A is the smallest value on the entire curve.

On the other hand, points B and C are global maxima since they share the largest value of the entire function.

Maxima and Minima

A curve has local maxima or minima when the curve changes y-direction. This means that a local maximum is greater than the points immediately to its left and right on the graph; likewise, a local minimum is greater than the points immediately to its left and right on the graph.

The global maxima and minima are the overall maximum or minimum points on the curve.

This can be done with a calculator using 2nd > CALC > maximum/minimum.

Exercise

Consider the function f(x)=x3−5x2+x−3.

On the curve of y=f(x), find, to three significant figures, the coordinates of

the local maxima,
the local minima.

Select the correct option

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Even functions

An even function is one for which

f(−x)=f(x) for all x∈R

Graphically, this means the function is symmetric in the y-axis:

Odd functions

An odd function is one for which

f(−x)=−f(x) for all x∈R

Graphically, this means the function has pointwise symmetry with the origin:

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