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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, $\text{}a\text{}$ (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.
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
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.
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.
Evaluate $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}+3x-4\text{\hspace{0.17em}}$ at
Replace the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the function with each specified value.
and we know that
Now we combine the results and simplify.
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