Functions
Function: A function is a relationship between two variables (one is called independent
variable, another is called dependent variable) such that to each value of the independent
variable there corresponds exactly one value of the dependent variable.
• Domain of the function y = f (x): D = The set of all values of the independent variable
x for which the function is defined.
• Range of the function: R = The set of all values taking on by the dependent variable
y.
Four ways to represent functions:
• Verbally: by a description in words;
• numerically: by a table of values;
• visually: by a graph {(x, f (x))|x ∈ D};
• algebraically: by an explicit formula.
Example 1 f (x) =
Example 2 f (x) =
x2
x2 −3x+2
√
is a function, D = {x : x 6= 1, 2}.
4 − x2 is a function, D = {x : −2 ≤ x ≤ 2}.
Example 3 f (x) = ±x2 is not a function.
(
Example 4 Piecewise defined functions: f (x) =
2x, x ≤ 0;
3x, x > 0.
Example 5 Evaluate a function: Let f (x) = x2 − 2x + 1. Then f (5) = 52 − 2(5) + 1 = 16.
f (x + 3) = (x + 3)2 − 2(x + 3) + 1 = x2 + 4x + 4.
Basic functions such as:
• Power function: f (x) = xc , where c is a real number.
• Polynomials: P (x) = an xn + an−1 xn−1 + ... + a1 x + a0 , where n is a positive integer
(which is called the degree of P (x)).
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• Rational functions:
P (x)
,
Q(x)
where P (x) and Q(x) are polynomials.
Some important functions:
• One-to-one function: y = f (x) is 1-1 ⇔ for each y ∈ R, there is only one x ∈ D.
Horizontal line test can be used to check this.
Example 6 f (x) = x2 is not 1-1; g(x) =
√
x + 5 is 1-1.
• Composite function f (g(x)) or f ◦g(x). The domain of f ◦g(x) is the set of all x ∈ D(g)
such that g(x) ∈ D(f ).
Example 7 Let f (x) =
√
2x − 4, g(x) =
√
3 − 2x.
p √
= {x ≤ −0.5} (the solution of 3−2x ≥ 0
Then f (g(x)) = 2 3 − 2x − 4 with
pD(f ◦g)
√
√
and 2 3 − 2x − 4 ≥ 0). g(f (x)) = 3 − 2 2x − 4, D(g ◦ f ) = {2 ≤ x ≤ 3.125} (the
√
solution of 2x − 4 ≥ 0 and 3 − 2 2x − 4 ≥ 0).
• Inverse function: y = f (x) → x = f −1 (y). We write it as y = f −1 (x).
– The graph of f −1 and the graph of f are symmetric about the line y = x.
– Cancelation: f (f −1 (y)) = y, f −1 (f (x)) = x.
To find inverse, we have two ways:
• Method 1: Using the coordinate geometry;
• Method 2: Using variable interchange.
Example 8 Let f (x) =
3x+2
,
5x−4
find the inverse f −1 (x).
Strategy to use variable interchange:
1) Write y = 3x+2
;
5x−4
3y+2
2) Switch x and y: x = 5y−4
;
4x+2
3) Isolate y: y = 5x−3 ;
4) Answer: y = f −1 (x) = 4x+2
.
5x−3
3
Example. let f (x) = 2x + 1, find the inverse f −1 (x).
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