2.3: Function Notation

It is common to let a letter denote a set of ordered pairs that is a function.

Note

If F denotes a function, then for any ordered pair, (x, y) in F, y is denoted by F(x).

Thus the ordered pair (x, F(x)) is an ordered pair of F. The notation makes it very easy to describe a function by an equation. Instead of

'Let \(F\) be the collection of ordered number pairs to which an ordered pair \((x, y)\) belongs if and only if \(x\) is a number and \(y = x^{2} + x\).'

one may write

'Let \(F\) be the function such that for all numbers \(x\), \(F(x) = x^{2} + x\).'

Operationally, you will find that often you can simply replace y in an equation by F(x) and define a function. Furthermore, because you are used to using y in an equation, you can often replace F(x) in the definition of a function by y and use the resulting equation which is more familiar.

The following expressions all may be used in the definition of a function.

1. \(F(x)=x^{2}\)
2. \(F(x)=x+5\)
3. \(F(x)=x+\frac{1}{x} \quad x \neq 0\)
4. \(F(x)=e^{x}\)
5. \(F(x)=\sqrt{x} \quad x \geq 0\)
6. \(F(x)=\log _{10} x \quad x>0\)

Observe that some conditions, \(x \neq 0\) and \(x \geq 0\) and \(x > 0\), are included for some of the expressions. Those conditions describe the domain of the function. For example, the domain of the function f defined by \(F(x) = log_{10} x\) is the set of positive numbers.

You may also see something like

\[\begin{array}{ll}
F(x)=\sqrt{1+x} & F(x)=\frac{1-x}{1+x} \\
F(x)=\sqrt{1-x^{2}} & F(x)=\log _{10}\left(x^{2}-x\right)
\end{array}\]

The intention is that the domain is the set of all values of x for which the expressions can be computed even though no restrictions are written. Often the restrictions are based on these rules:

Note

1. Avoid dividing by zero.
2. Avoid computing the square root of negative numbers.
3. Avoid computing the logarithm of 0 and negative numbers.

Assuming we use only real numbers and not complex numbers, complete descriptions of the previous functions would be

\[\begin{array}{llllll}
f(x) & = \sqrt{1+x}, & x \geq-1 \quad & \quad F(x) & = \frac{1-x}{1+x}, & x \neq-1 \\
F(x) & = \sqrt{1-x^{2}}, & -1 \leq x \leq 1 \quad & \quad F(x) & = \log _{10}\left(x^{2}-x\right), & x<0 & \text { or } \quad 1<x
\end{array}\]

Use of parentheses. The use of parentheses in the function notation is special to functions and does not mean multiplication. The symbol inside the parentheses is always the independent variable, a member of the domain, and \(F(x)\) is a value of the dependent variable, a member of the range. It is particularly tricky in that we will often need to use the symbol \(F(x + h)\), and students confuse this with a multiplication and replace it with \(F(x) + F(h)\). Seldom is this correct.

Example 2.3.1 For the function, R, defined by

\[\begin{array}{c}
R(x)=x+\frac{1}{x} \quad x \neq 0 \\
R(1+3)=R(4)=4+\frac{1}{4}=4.25 \\
R(1)=1+\frac{1}{1}=2.0 \quad \text { and } \\
R(3)=3+\frac{1}{3}=3.3333 \cdots \\
R(1)+R(3)=2+3.3333 \cdots=5.3333 \cdots \neq 4.25=R(4)
\end{array}\]

In this case

\[R(1+3) \neq R(1)+R(3)\]

Exercises for Section 2.3 Function Notation.

Exercise 2.3.1 Let \(F\) be the collection of ordered number pairs to which an ordered pair \((x, y)\) belongs if and only if \(x\) is a number and \(y = x^{2} + x\).

1. Which of the ordered number pairs belong to F? (0,1), (0,0), (1,1), (1,3), (1,-1), (-1,1), (-1,0), (-1,-1).
2. Is there any uncertainty as to the members of F?
3. What is the domain of F?
4. What is the range of F?

Exercise 2.3.2 For the function, \(F\), defined by \(F(x) = x^2\),

1. Compute \(F(1 + 2)\), and \(F(1) + F(2)\). Is \(F(1 + 2) = F(1) + F(2)\)?
2. Compute \(F(3 + 5)\), and \(F(3) + F(5)\). Is \(F(3 + 5) = F(3) + F(5)\)?
3. Compute \(F(0 + 4)\), and \(F(0) + F(4)\). Is \(F(0 + 4) = F(0) + F(4)\)?

Exercise 2.3.3 Find a function, \(L\), defined for all numbers (domain is all numbers) such that for all numbers \(a\) and \(b\), \(L(a + b) = L(a) + L(b)\. Is there another such function?

Exercise 2.3.4 Find a function, \(M\), defined for all numbers (domain is all numbers) such that for all numbers \(a\) and \(b\), \(M(a + b) = M(a) \times M(b)\). Is there another such function?

Exercise 2.3.5 For the function, \(F(x) = x^{2} + x\), compute the following

1. \(\frac{F(5)-F(3)}{5-3}\)
2. \(\frac{F(3+2)-F(3)}{2}\)
3. \(\frac{F(b)-F(a)}{b-a}\)
4. \(\frac{F(a+h)-F(a)}{h}\)

Exercise 2.3.6 Repeat steps (a) - (d) of Exercise 2.3.5 for the functions

1. \(F(x)=3 x\)
2. \(F(x)=x^{3}\)
3. \(F(x)=2^{x}\)
4. \(F(x)=\sin x\)

Exercise 2.3.7

1. In Figure 2.3.7A is the graph of \(y^{4} = x^{2}\) for \(−2 \leq x \leq 2\). Write equations that define five different maximal simple subgraphs.
2. In Figure 2.3.7B is the graph of \(|x| + |y| = 1\) for \(−1 \leq x \leq 1\). Write equations that define five different maximal simple subgraphs.

Figure for Exercise 2.3.7 Graph of A \(y^{4} = x^{2}\), and B \(|x| + |y| = 1\), for Exercise 2.3.7.

Exercise 2.3.8 What are the implied domains of the functions

\[\begin{array}{ll}
F(x)=\sqrt{x-1} & F(x)=\frac{1+x^{2}}{1-x^{2}} \\
F(x)=\sqrt{4-x^{2}} & F(x)=\log _{10}\left(x^{2}\right)
\end{array}\]