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Function Theory
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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A function from x to y is a special type of relation where each x value has only one possible y-value. It is expressed in the form
where f and x can be replaced by any letters.
A function can be evaluated for specific values of x by plugging the value into the expression of the function.
The domain of a function is the set of possible inputs it can be given.
The "natural" or "largest possible" domain of a function is all the values of x for which the expression f(x) is defined.
The range of a function is the set of possible values it can output.
If the domain of the function is restricted, the range may need to be restricted as a consequence.
The domain and range of functions are commonly intervals of real numbers.
For example, if f(x) is defined for 1<x≤5, we can write the domain
(∈ means "in" or "element of", and R is all real numbers)
We can also use the equivalent interval notation
where, by IB convention, an outward facing [ means that end is not inclusive (1<x) and an inward facing ] means that end is inclusive (x≤5).
Another common interval notation is
where ( indicates a non-inclusive endpoint and ] indicates an inclusive endpoint. In this style, all brackets are inward facing.
These can also be visualized on a number line:
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A function as a model means using a mathematical relationship to represent real-world phenomena.
By assigning input values (independent variables) and calculating corresponding outputs (dependent variables), a function allows us to approximate, describe, or predict patterns, behaviors, or outcomes.
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.
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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:
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We can find the value f(a) of a function by looking at the y-value of the graph where x=a:
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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.
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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.
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.
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.
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).
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).
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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.
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.
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Vertical asymptotes have the equation x=a for some a∈R.
Vertical asymptotes occur for functions of the form
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.
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.
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Graphing calculators can be used to find the coordinates of maxima and minima of a function. On a TI-84, click 2nd > CALC and then either 3:minimum or 4:maximum.
Before trying to find a minimum or maximum, check visually whether the function has the desired extrema.
An even function is one for which
Graphically, this means the function is symmetric in the y-axis:
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An odd function is one for which
Graphically, this means the function has pointwise symmetry with the origin:
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Functions can be composed by passing the output of one into the other. We use the symbol ∘, and pay close attention to the order in which functions are composed:
To find an expression for f(g(x)), substitute g(x) for x in the expression for f(x).
We can find x=f−1(b) by applying the function to both sides:
So finding f−1(b) is equivalent to solving f(x)=b.
Graphically, find f−1(b) is equivalent to being given y=b, and finding the value of x for which that is true:
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The graph of a function f shows all the points (x,f(x)). Since f−1 undoes f, its graph will show all the points (f(x),x). In other words, the x and y values are swapped.
This is equivalent to reflecting the curve y=f(x) in the line y=x:
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Since f−1 undoes f, the domain of f−1 is all the possible values f could output. That is, the domain of f−1 is the range of f.
The range of f−1 is all the possible values that could have gone into f. Thus, the range of f−1 is the domain of f.
Formally, the inverse function is such that
We call x the identity function, as I(x)=x composed with any function gives the same function.
Inverse functions f−1(x) can be found algebraically by switching x and y in the expression for f(x) and attempting to isolate y.
Recall that a function f must pass the vertical line test to guarantee that each input gives at most one output.
Since f−1 is also a function, it too must pass the vertical line test. But since the graph of f−1(x) is a reflection of the graph of f(x) in the line y=x, the graph of f(x) must then pass the horizontal line test:
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A function that passes the horizontal line test is said to be one to one - each input yields exactly one distinct output. Such a function is said to be invertible, which means f−1 exists.
A function is said to be self-inverse if it is its own inverse:
Since the graph of f−1(x) is the mirror image of f(x) in the line y=x, the graph of a self-inverse function must be symmetric in the line y=x.
When a function f is not one to one, ie it does not pass the horizontal line test, its domain needs to be restricted for the inverse to exist.