# 3.1 Functions and function notation  (Page 4/21)

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 $n$ 1 2 3 4 5 $Q$ 8 6 7 6 8

[link] displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

 Age in years, (input) 5 5 6 7 8 9 10 Height in inches, (output) 40 42 44 47 50 52 54

Given a table of input and output values, determine whether the table represents a function.

1. Identify the input and output values.
2. Check to see if each input value is paired with only one output value. If so, the table represents a function.

## Identifying tables that represent functions

Which table, [link] , [link] , or [link] , represents a function (if any)?

Input Output
2 1
5 3
8 6
Input Output
–3 5
0 1
4 5
Input Output
1 0
5 2
5 4

[link] and [link] define functions. In both, each input value corresponds to exactly one output value. [link] does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by [link] can be represented by writing

Similarly, the statements

represent the function in [link] .

[link] cannot be expressed in a similar way because it does not represent a function.

Does [link] represent a function?

Input Output
1 10
2 100
3 1000

yes

## Finding input and output values of a function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

## Evaluation of functions in algebraic forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function $\text{\hspace{0.17em}}f\left(x\right)=5-3{x}^{2}\text{\hspace{0.17em}}$ can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

Given the formula for a function, evaluate.

1. Replace the input variable in the formula with the value provided.
2. Calculate the result.

## Evaluating functions at specific values

Evaluate $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}+3x-4\text{\hspace{0.17em}}$ at

1. $2$
2. $a$
3. $a+h$
4. $\frac{f\left(a+h\right)-f\left(a\right)}{h}$

Replace the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the function with each specified value.

1. Because the input value is a number, 2, we can use simple algebra to simplify.
$\begin{array}{ccc}\hfill f\left(2\right)& =& {2}^{2}+3\left(2\right)-4\\ & =& 4+6-4\hfill \\ & =& 6\hfill \end{array}$
2. In this case, the input value is a letter so we cannot simplify the answer any further.
$f\left(a\right)={a}^{2}+3a-4$
3. With an input value of $\text{\hspace{0.17em}}a+h,\text{\hspace{0.17em}}$ we must use the distributive property.
$\begin{array}{ccc}\hfill f\left(a+h\right)& =& {\left(a+h\right)}^{2}+3\left(a+h\right)-4\hfill \\ & =& {a}^{2}+2ah+{h}^{2}+3a+3h-4\hfill \end{array}$
4. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that
$f\left(a+h\right)={a}^{2}+2ah+{h}^{2}+3a+3h-4$

and we know that

$f\left(a\right)={a}^{2}+3a-4$

Now we combine the results and simplify.

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