Functions

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What is a Function?

A function is a special relationship where each input has a single output. It is often written as \(f(x)\) where x is the input value. Example: \(f(x) = \frac{x}{2}\) ("f of x equals x divided by 2"). It is a function because each input \(x\) has a single output.

Vertical Line test

Image result for vertical line test for functions diagram

A test that can be used to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point. Note that all functions are relationships, but not all relationships are functions.

Domain and Range

If you are comfortable with domain and Range already in forms like  Domain X ∈ [1,5] or 1 ≤ X ≤ 5,  Range Y ∈ ]2,10[ or 2 < X < 5   Then you can move on.

If you have any issues this might help:

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Composite Functions

The composition of two functions \(f(x)\) and \(g(x)\) such that function g(x) is applied first and function f(x) is applied second is the function fg(x). It maps x to f(g(x)) and we write:

(fg)(x)=f(g(x))

Be careful to read composite functions in the correct order, read from right to left. The function closest to the x is the first function to be executed, the result of which is given to the next.

The domain of composite functions

The domain of the function f∘g is the set of all possible x values of g such that the respective g(x) belongs to the domain of f.

Inverse functions

Now let us consider the function f(x)=2x.

f-1(x) undoes the action of f(x) and therefore f-1(x)=x/2.

It is enormously important that you do not think of it as "one over f". Note that 1f(x) is written as [f(x)-1 ]

On a graph, f(x) and  f1 are symmetric in the line y = x, meaning that if you reflect f(x) in y=x then you obtain f1, and vice versa.

Finding the inverse of functions ( f-1(x))

  1. Write y=f(x).
  2. Then replace all the x's with y's, and the y's by x's.
  3. Solve until you obtain an expression y=g(x), where g(x) includes all components except y.

 

The Reciprocal function

The reciprocal function of f(x) is the function that maps any x value (except x≠0) to 1x and thus,
 
f(x)=1x
 
Remember the reciprocal function and any multiple is self-inverse (the inverse is the same as the original).

 

Transformations of Functions and Graphs

There are 3 types of transformations:

Translations

Variablefunctionresultnotes
hhy=f(x-h)y=f(x-h)Translating the graph towards the right/leftRight when hh is positive
vvy=f(x)+vy=f(x)+vTranslating the graph upwards/downwardsUpwards when vv is positive

\graph{{x^2+1},{(x-1)^2}}

Scalings

Variablefunctionresultnotes
ssy=s(f(x))y=s(f(x))Stretching/compressing the graph along the y-axisStretching when s > 1
ccy=f(cx)y=f(cx)Compressing/stretching the graph along the x-axisCompressing when c > 1

\graph{{1.5x^2},{0.2x^2}}

Reflections

functionresult
y=-f(x)Reflection across y-axis
y=f(-x)Reflection across x-axis

\graph{{y=x}}

Review

Type of TransformationEquation
Vertical Translationy=f(x)+vy=f(x)+v
Horizontal Translationy=f(x-h)y=f(x-h)
Vertical Stretchy=sf(x) y=sf(x)
Horizontal Stretchy=f(cx) y=f(cx)
Vertical Reflectiony=-f(x) y=-f(x)
Horizontal Reflectiony=f(-x) y=f(-x)

It is also worth noting that more than one transformation can occur. For example, the function y=2x2+3y=2x^2+3 can be viewed as the basic quadratic function y=x2y=x^2 stretched vertically by a factor of 2 and shifted vertically by a factor of 3.

\graph{{4x^2+2}}

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