FUNCTION FACTS
Definition of a Function:
A function is a rule that describes how one quantity depends upon another.
• 𝑓𝑓(𝑥𝑥)=𝑦𝑦 is read “𝑓𝑓 of x.”
• The output variable, 𝑦𝑦 is the dependent variable because it depends on the input variable, x
which is called the independent variable.
• For each input x, there is only one possible output y.
Example: The set of points {(1,2), (2,4), (3,-1), (4,4)} is a function.
The set of points {(1,2), (2,4), (3,-1), (3,4)} is not a function since an input of 3
yields more than one output.
Vertical Line Test: This tests whether or not a relation between two variables is a function.
If a vertical line crosses the curve more than once, the relation is not a function.
Domain: The domain is the set of all possible values of x for which the function 𝑓𝑓(𝑥𝑥) exists.
• x cannot cause a denominator to be zero.
• If x is under a square root (or any even root) sign, x cannot cause the expression under the
root sign to be negative (when using real numbers).
• x must be greater than 0 for 𝑦𝑦=log𝑏𝑏𝑥𝑥 .
Range: The range is the set of all possible values of the function, that is, the output variable, 𝑦𝑦.
Values of Functions:
Example: Let 𝑓𝑓(𝑥𝑥)=𝑥𝑥2+ 4𝑥𝑥 − 3 .
Find 𝑓𝑓(2): 𝑓𝑓(2)= 22+ 4(2)−3 = 4 + 8 −3 = 9
Find 𝑓𝑓(𝑥𝑥+ 1): 𝑓𝑓(𝑥𝑥+ 1)= (𝑥𝑥+ 1)2+ 4(𝑥𝑥+ 1)−3 = 𝑥𝑥2+ 6𝑥𝑥+ 2
Algebra of Functions:
Sum:
(𝑓𝑓+𝑔𝑔)(𝑥𝑥)=𝑓𝑓(𝑥𝑥)+𝑔𝑔(𝑥𝑥)
Difference:
(𝑓𝑓 − 𝑔𝑔)(𝑥𝑥)=𝑓𝑓(𝑥𝑥)− 𝑔𝑔(𝑥𝑥)
Product:
(𝑓𝑓𝑔𝑔)(𝑥𝑥)=𝑓𝑓(𝑥𝑥)∙ 𝑔𝑔(𝑥𝑥)
Quotient:
�𝑓𝑓
𝑔𝑔�(𝑥𝑥)=𝑓𝑓(𝑥𝑥)
𝑔𝑔(𝑥𝑥)
for
𝑔𝑔(𝑥𝑥)≠0
This is a function.
This is not a function.
