When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\left(x\right)=5 - 3^[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.
Given the function [latex]h\left(p\right)=
^+2p[/latex], evaluate [latex]h\left(4\right)[/latex].
Show SolutionTo evaluate [latex]h\left(4\right)[/latex], we substitute the value 4 for the input variable [latex]p[/latex] in the given function.
Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[/latex].
In the following video we offer more examples of evaluating a function for specific x values.
For the function, [latex]f\left(x\right)=^+3x - 4[/latex], evaluate each of the following.
Replace the [latex]x[/latex] in the function with each specified value.
Given the function [latex]g\left(m\right)=\sqrt[/latex], evaluate [latex]g\left(5\right)[/latex].
Show SolutionIn the next video, we provide another example of how to solve for a function value.
Given the function [latex]h\left(p\right)=
^+2p[/latex], solve for [latex]h\left(p\right)=3[/latex].
Show SolutionIf [latex]\left(p+3\right)\left(p - 1\right)=0[/latex], either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p - 1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.
[latex]\begin&p+3=0, &&p=-3 \\ &p - 1=0, &&p=1\hfill \end[/latex]
This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3[/latex].
parabola with labeled points (-3, 3), (1, 3), and (4, 24)." width="487" height="459" />
We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24[/latex].
Given the function [latex]g\left(m\right)=\sqrt[/latex], solve [latex]g\left(m\right)=2[/latex].
Show SolutionSome functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p[/latex]. We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n[/latex].
Express the relationship [latex]2n+6p=12[/latex] as a function [latex]p=f\left(n\right)[/latex], if possible.
Show SolutionTo express the relationship in this form, we need to be able to write the relationship where [latex]p[/latex] is a function of [latex]n[/latex], which means writing it as [latex]p=[/latex] expression involving [latex]n[/latex].
Therefore, [latex]p[/latex] as a function of [latex]n[/latex] is written as
It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.
Watch this video to see another example of how to express an equation as a function.
Does the equation [latex]^+^=1[/latex] represent a function with [latex]x[/latex] as input and [latex]y[/latex] as output? If so, express the relationship as a function [latex]y=f\left(x\right)[/latex].
Show SolutionFirst we subtract [latex]^[/latex] from both sides.
We now try to solve for [latex]y[/latex] in this equation.
We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\left(x\right)[/latex].
If [latex]x - 8^=0[/latex], express [latex]y[/latex] as a function of [latex]x[/latex].
Show Solution [latex]y=f\left(x\right)=\cfrac>[/latex]\sqrt[3]Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?
Yes, this can happen. For example, given the equation [latex]x=y+^[/latex], if we want to express [latex]y[/latex] as a function of [latex]x[/latex], there is no simple algebraic formula involving only [latex]x[/latex] that equals [latex]y[/latex]. However, each [latex]x[/latex] does determine a unique value for [latex]y[/latex], and there are mathematical procedures by which [latex]y[/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[/latex] as a function of [latex]x[/latex], even though the formula cannot be written explicitly.
As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.
Pet | Memory span in hours |
---|---|
Puppy | 0.008 |
Adult dog | 0.083 |
Cat | 16 |
Goldfish | 2160 |
Beta fish | 3600 |
At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[/latex].
The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function [latex]P[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text\right)=2160[/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.
Using the table below,
[latex]n[/latex] | 1 | 2 | 3 | 4 | 5 |
[latex]g(n)[/latex] | 8 | 6 | 7 | 6 | 8 |
[latex]n[/latex] | 1 | 2 | 3 | 4 | 5 |
[latex]g(n)[/latex] | 8 | 6 | 7 | 6 | 8 |
When we input 2 into the function [latex]g[/latex], our output is 6. When we input 4 into the function [latex]g[/latex], our output is also 6.
Using the table from the previous example, evaluate [latex]g\left(1\right)[/latex] .
Show Solution [latex]g\left(1\right)=8[/latex]Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).
Given the graph below,
Using the graph, solve [latex]f\left(x\right)=1[/latex].