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. Show
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 FormsWhen we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]\text{}f\left(x\right)=5-3{x}^{2}\text{}[/latex] 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.
Evaluating Functions at Specific Values Evaluate [latex]\text{}f\left(x\right)={x}^{2}+3x-4\text{}[/latex] at
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Solution Evaluating Functions Expressed in FormulasSome 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]\text{}2n+6p=12\text{}[/latex] expresses a functional relationship between [latex]\text{}n\text{}[/latex] and [latex]\text{}p\text{}[/latex]. We can rewrite it to decide if [latex]\text{}p\text{}[/latex] is a function of [latex]\text{}n\text{}[/latex]. Given a function in equation form, write its algebraic formula.
Express the relationship [latex]\text{}2n+6p=12\text{}[/latex] as a function [latex]\text{}p=f\left(n\right)\text{}[/latex], if possible. AnalysisIt is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula. Solution
Solution 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]\text{}x=y+{2}^{y},\text{}[/latex] if we want to express [latex]\text{}y\text{}[/latex] as a function of [latex]\text{}x,\text{}[/latex] there is no simple algebraic formula involving only [latex]\text{}x\text{}[/latex] that equals [latex]\text{}y\text{}[/latex]. However, each [latex]\text{}x\text{}[/latex] does determine a unique value for [latex]\text{}y\text{}[/latex], and there are mathematical procedures by which [latex]\text{}y\text{}[/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]\text{}y\text{}[/latex] as a function of [latex]\text{}x\text{}[/latex], even though the formula cannot be written explicitly. Evaluating a Function Given in Tabular FormAs 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 Table 1-10.[1] Table 1-10
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]\text{}P\text{}[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text{goldfish}\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]\text{}P\text{}[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form. Given a function represented by a table, identify specific output and input values.
Using Table 1-11,
Solution Finding Function Values from a GraphEvaluating 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 in Figure 1-4,
Solution Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions What is a function called when its return value is always the same for the same arguments?In computer programming, a pure function is a function that has the following properties: the function return values are identical for identical arguments (no variation with local static variables, non-local variables, mutable reference arguments or input streams), and.
Can functions have same output?Each input has only one output. Each input has only one output, and the fact that it is the same output (4) does not matter. This relation is a function. Remember that in a function, the input value must have one and only one value for the output.
What is the output of a function called?The set of input values is called the domain of the function. And the set of output values is called the range of the function.
What is pure function with example?A function is called pure function if it always returns the same result for same argument values and it has no side effects like modifying an argument (or global variable) or outputting something. The only result of calling a pure function is the return value. Examples of pure functions are strlen(), pow(), sqrt() etc.
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