Section 2.1: Functions A function f from a set A into a set B is an association between elements in A and B where for each element in A there is exactly one element in B. The set A is called the domain of f and the B is called the range of f . It may be useful to think of a function in terms of set diagrams.
Is f a function?
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Is g a function?
Is h a function?
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Perhaps the most common way to represent functions in this course is symbolically in functional notation. For a function f which may be denoted symbolically, the standard functional notation is given by f (x) = y, where y is some symbolic expression which assigns any input x to exactly one output y.
Let f be defined by f (x) = x2 . If we put any real value in for x, then there is only one output. For example, f (2) has but one value, i.e., f (2) = 22 = 4. Thus, f is a function since for each input there is a unique output.
Exercises: Use the function to evaluate the indicated expressions and simplify. i) f (x) = x2 − 2x
ia) f (1)
ib) f (3)
ic) f (−x)
id) f ( x1 )
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ii) g(t) = t −
1 t
iia) g(1)
iib) g
(1) a
iic) g(0)
3x − 1, if x < 0 x2 , if 0 ≤ x < 4 iii) h(x) = 16, if x ≥ 4 iiia) h(3)
iiib) h(0)
iiic) h(27)
iiid) h(2t), where t < 0
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Exercises: Find f (a), f (a + h), and the difference quotient Simplify the difference quotient.
f (a + h) − f (a) , with h ̸= 0. h
i) f (x) = x2 − 8
ii) f (x) = 101
iii) f (x) =
1 x+1
Note: Difference quotients are of paramount importance in differential calculus.
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It will often be the case that the real-valued functions studied in this course will have a domain of all real numbers. However, there are special cases where the domain must be restricted since all real numbers cannot always be inputs. We call the domain we find for a symbolic function the implied domain. Special Cases with Restrictions on Domain 1) Functions with variables under even indexed radicals √ Consider, for example, f (x) = x + 6. Not every real number can be an input. Suppose we try√−7 as an input. √ √ Well, f (−7) = −7 + 6 = −1; but −1 is not a real number. Since inputting −7 gives an invalid output, then −7 is not a valid input. √ The question, then, is this: How do we find the valid inputs or the Domain of x + 6? We consider the restriction on the inputs. Since x + 6 cannot be negative, then x + 6 ≥ 0. We solve this inequality to find what x must at least be. x + 6 ≥ 0 =⇒ x ≥ −6. So, our restriction is x ≥ −6. Thus, Domain f = {x|x ≥ −6} [i.e., all real numbers greater than or equal to −6].
2) Functions with variables in denominator 1 Consider, for example, g(x) = x+7 . Not every real number can be an input. What can go wrong here? Since there is a variable in the denominator, the denominator could be zero [precisely when −7 is the input as we will see]. Recall that division by zero is not a valid operation. So, our restriction is that the denominator cannot be zero. That is, x+7 ̸= 0 =⇒ x ̸= −7.
Thus, Domain g = {x|x ̸= −7} [i.e., all real numbers except −7].
Examples: Find the implied domain of each function. i) f (x) =
√
18 − 3x
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ii) g(x) =
12x x2 − 9
11x + 5 iii) h(x) = √ x−1
√ iv) k(x) =
x2 − 3x x−4
Additional Practice Pages 155-157 vocabulary 11-29 odd, 32, 35, 36, 38, 43, 49, 62, 67
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