From Three Inputs to One Output
A quantity may depend on three different inputs at the same time. For example, a total might depend on the amounts of three different products, or a geometric measurement might depend on the three dimensions of a box. A mathematical model describes how those inputs determine an output.
The key is to make each input’s role explicit. Once the roles are fixed, an ordered triple supplies one value for each input, and the function assigns that triple a single output.
The three inputs are written in a fixed order: first \(x\), then \(y\), then \(z\). Thus \(f(2,5,3)\) means that \(x=2\), \(y=5\), and \(z=3\), in that order. The variables can have different meanings, and their values cannot be reassigned to different positions without changing the input.
How a Verbal Model Becomes a Function
To translate a description into a function, first identify the output being modeled. Then identify the three inputs and state what each represents. Finally, express the stated relationship between those inputs and the output as a formula.
For example, if a total consists of a fixed amount plus three separate contributions, the formula should include the fixed amount and one term for each contribution. If a description says that two inputs are multiplied, the corresponding term must include their product. The formula should reflect the relationships stated, rather than simply placing three variables into an expression that looks plausible.
Decide which single quantity the function is meant to produce.
State what the first, second, and third variables represent, in order.
Write a formula that includes the contributions or operations specified in the description.
Substitute the given triple into the function in its stated order and simplify.
Worked Example: A Total from Three Contributions
Worked Example: A Total from Three Contributions
A model has a fixed contribution of \(80\). Each unit of the first input contributes \(7\), each unit of the second contributes \(4\), and each unit of the third contributes \(9\). Let \(x,y,z\) denote the amounts of the first, second, and third inputs. Write a function for the total, then find its value at \((2,5,3)\).
The output is the total. The fixed contribution is \(80\), and the three input-dependent contributions are \(7x\), \(4y\), and \(9z\). Therefore the function is
In the ordered triple \((2,5,3)\), the first input is \(2\), the second is \(5\), and the third is \(3\). Substitute those values in the corresponding positions:
The three input-dependent contributions add to \(14+20+27=61\), and adding the fixed contribution gives \(80+61=141\). This checks the result by grouping the terms according to their roles in the model.
Why the Formula Represents the Description
For a model made from separate contributions, the formula follows by adding those contributions. Suppose a starting amount is \(A\), and the three inputs contribute \(r_1x\), \(r_2y\), and \(r_3z\), respectively. The total is the starting amount together with all three contributions, so
This construction works because each term records one part of the stated total. For a fixed triple, the contributions are numbers, and their sum is one number. The formula therefore defines a scalar output for that input triple. In the worked example, \(A=80\), \(r_1=7\), \(r_2=4\), and \(r_3=9\), giving exactly \(F(x,y,z)=80+7x+4y+9z\).
This reasoning also shows why the meanings assigned to the variables matter. The coefficient \(7\) belongs with \(x\) because the description assigns that contribution to the first input. Pairing \(7\) with another variable would describe a different model.
Worked Example: Surface Area from Three Dimensions
Worked Example: Surface Area from Three Dimensions
Let \(l\), \(w\), and \(h\) denote the length, width, and height of a rectangular box. Write a function for its surface area and evaluate it when \(l=2\), \(w=3\), and \(h=4\).
A rectangular box has two faces with area \(lw\), two with area \(lh\), and two with area \(wh\). Adding the areas of all six faces gives
Now substitute the triple \((2,3,4)\), keeping its coordinates matched to \(l,w,h\):
To check, the three distinct face areas are \(2\cdot3=6\), \(2\cdot4=8\), and \(3\cdot4=12\). Each occurs twice, so the total is \(2(6)+2(8)+2(12)=12+16+24=52\), as before.
Not Every Three-Input Model Is Additive
Some models combine inputs by multiplication or powers rather than by adding separate contributions. The function notation is the same: define what each input means, state the relationship in a formula, and then evaluate the formula at the requested triple.
Worked Example: A Model with a Product and a Square
Suppose an index \(G\) is defined as twice the product of its first two inputs, plus the square of its third input. Let the inputs be \(a,b,c\). Write the function and evaluate it at \((2,3,4)\).
The product of the first two inputs is \(ab\). Twice that product is \(2ab\), and the square of the third input is \(c^2\). The function is therefore
Substituting \(a=2\), \(b=3\), and \(c=4\) gives
Check the two parts separately: the first two inputs have product \(2\cdot3=6\), and twice that is \(12\); the third input has square \(4^2=16\). Their sum is \(12+16=28\), confirming the output.
Input Order and a Common Mistake
A function’s formula and its input order work together. In \(F(x,y,z)=80+7x+4y+9z\), the first coordinate supplies the value of \(x\), the second supplies \(y\), and the third supplies \(z\). Changing which coordinate goes with which variable changes the evaluation.
For instance, a mistaken evaluation of the first example might swap the second and third coordinates, using \((2,3,5)\) in place of \((2,5,3)\):
That calculation is arithmetically consistent, but it evaluates \(F(2,3,5)\), not \(F(2,5,3)\). For the requested triple, the second coordinate is \(5\) and the third is \(3\), so the correct substitution is
A reliable check is to write the variable assignments explicitly before substituting: \(x=2\), \(y=5\), and \(z=3\). This small step helps prevent an order error even when the formula itself is correct.
Why Three-Input Models Are Useful
Writing a model as a function separates the general relationship from any one calculation. The formula \(F(x,y,z)=80+7x+4y+9z\) describes the total for every input triple to which the model applies, rather than only for the triple \((2,5,3)\). A new triple can be evaluated using the same rule.
Function notation also makes the roles of inputs visible. A formula records which quantities combine additively, which are multiplied, and which are raised to powers. That structure is useful when comparing different input combinations or when communicating exactly how a model produces its output.
Before using a three-input model, check that the variables have been assigned clearly, that the formula matches the stated relationship, and that each coordinate is substituted into the correct position. These checks address the central challenges in translating and evaluating such models.
Check Your Understanding
For each question, identify the input roles before carrying out any calculation.
- A function is \(H(x,y,z)=5x+2y+3z\). Evaluate \(H(1,4,2)\), showing each substitution.
- A total is a fixed amount of \(30\), plus \(6\) times the first input, plus \(2\) times the second input, plus \(5\) times the third input. Write the function.
- For \(P(a,b,c)=ab+c^2\), evaluate \(P(3,2,5)\) and check the product and square separately.
- In \(F(x,y,z)=80+7x+4y+9z\), what function value is computed by the expression \(80+7(2)+4(3)+9(5)\)?
- Explain why a formula for a scalar function of three variables must assign one output to each input triple in its domain.