Real-valued functions

1. Isabel Silva Magalhães Fundamentals of real-valued functions with real variables
Real-valued functions with real variables
Definitions and Notation
A function f is a rule that assigns to each element x in a set D exactly one element,
called f(x), in a set E.
f :
D −→ E
x 7−→ f(x)
D and E are sets of real numbers (D ⊆ R and E ⊆ R)
D : domain of f
E : codomain of f
f(x) : value of f at x or image of x under f
range(f) = {f(x) : x ∈ D}
graph(f) = {(x,y) ∈ R2 : x ∈ D,y = f(x)}
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2. Isabel Silva Magalhães Fundamentals of real-valued functions with real variables
Real-valued functions with real variables
Definitions and Notation
Even function: f(x) = f(−x),x ∈ D (symmetric about the y-axis)
Odd function: f(x) = −f(−x),x ∈ D (symmetric about the origin)
Periodic function (period p): f(x) = f(x+p),x ∈ D (graph of f replicates itself
on intervals of length p)
One-to-one (or injective) function: f(x1) 6= f(x2) whenever x1 6= x2 (it never
takes on the same value twice)
I Horizontal line test: a function is one-to-one if and only if no horizontal line
intersects its graph more than once
Onto (or surjective) function: codomain of f = range of f
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3. Isabel Silva Magalhães Fundamentals of real-valued functions with real variables
Real-valued functions with real variables
Composition (or composite) of f and g: (“f circle g”)
f ◦g(x) = f(g(x))
Domain of f ◦g(x) : {x ∈ R : x ∈ Dg,g(x) ∈ Df }
Inverse function of f: f−1 (f inverse)
Let f be a one-to-one function with domain A and range B. Then its inverse
function, f−1, has domain B and range A and is defined by
f−1
(y) = x ⇔ f(x) = y,
for any in y ∈ B
f−1 ◦f(x) = f−1(f(x)) = x, for every x ∈ A
f ◦f−1(x) = f(f−1(x)) = x, for every x ∈ B
The graph of f−1 is obtained by reflecting the graph f of about the line y = x
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4. Isabel Silva Magalhães Fundamentals of real-valued functions with real variables
Real-valued functions with real variables
New functions from old functions
c Expression Geometrical transformation Example
c > 0 f(x)±c Vertical translation (upward/downward shift) x2 +1
c > 0 f(x±c) Horizontal translation (shift to the left/right) (x+1)2
c = −1 −f(x) Reflection about the x-axis −x2
c = −1 f(−x) Reflection about the y-axis e−x
c > 1 cf(x) Stretch the graph vertically by a factor of c 2sin(x)
c > 1 f(x)/c Compress the graph vertically by a factor of c sin(x)/2
c > 1 f(cx) Compress the graph horizontally by a factor of c sin(2x)
c > 1 f(x/c) Stretch the graph horizontally by a factor of c sin(x/2)
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