[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,
$\text{}a\text{}$ (input)
5
5
6
7
8
9
10
Height in inches,
$\text{}h\text{}$ (output)
40
42
44
47
50
52
54
Given a table of input and output values, determine whether the table represents a function.
Identify the input and output values.
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
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.
Replace the input variable in the formula with the value provided.
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
$2$
$a$
$a+h$
$\frac{f\left(a+h\right)-f\left(a\right)}{h}$
Replace the
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the function with each specified value.
Because the input value is a number, 2, we can use simple algebra to simplify.
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.