Inputing Values into a Function

In mathematics, we often assign numerical values to variables in equations or mathematical expressions. A function can be thought of as a machine: you input a value (called the input), the function processes it according to a specific rule, and produces a result (called the output). The input is usually represented by \(x)\, and the output by $f(x)$ or $y$ .

Understanding Variables and Substitution

A variable is a placeholder for an unknown number, often called the "unknown". When we replace a variable with a specific value, we refer to this process as substitution.

If an equation contains only one variable, we can solve for its value by isolating it.
If an equation contains multiple variables, there can be multiple solutions. Each value substituted affects the remaining variables in the equation.

When we talk about inputting values into a function, we are essentially substituting the variables in a mathematical expression or equation with specific numerical values to evaluate the result.
Often "solving an equation" is actually finding the values of the variables inside it.

For example

Suppose we have three variables, two of which have known values:

$X=3$

$Y=2$

$Z=\text{?}$

We are also given the equation:

$X^2+Y=Z$

Remember! when facing this kind of questions, you want to try and find the values of the variables.
To solve the problem, we will first substitute the known values into the equation:

$3^2+2=Z$

Simplify and solve:

$9+2=Z$

$Z=11$

By substituting the known values and performing the necessary operations, we were able to isolate and calculate the value of $Z$

Answer: $Z=11$

X Y Z By assigning the numerical value, the general form becomes a particular case

Detailed explanation

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Test yourself on domain of a function!

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

Practice more now

If you are interested in this article, you may also be interested in the following articles:

Graphical Representation of a Function

Algebraic Representation of a Function

Function Notation

Domain of a Function

Indefinite Integral

Variation of a Function

Increasing Function

Decreasing Function

Constant Function

Functions for Seventh Grade

Intervals of Increase and Decrease of a Function

Table of Values

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Examples and exercises with solutions for assigning numerical value in a function

Exercise #1

Does the given function have a domain? If so, what is it?

$\frac{9x}{4}$

Video Solution

Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers. This means that any one of the x values could be compatible.

Answer

No, the entire domain

Exercise #2

Look at the following function:

$\frac{5}{x}$

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

Since the unknown variable is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, $x\ne0$

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

Answer

Yes, $x\ne0$

Exercise #3

$\frac{6}{x+5}=1$

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To solve this problem, we will determine the domain, or field of application, of the equation $\frac{6}{x+5} = 1$ .

Step-by-step solution:

Step 1: Identify the denominator. In the given equation, the denominator is $x+5$ .
Step 2: Determine when the denominator is zero. Solve for $x$ by setting $x+5 = 0$ .
Step 3: Solve the equation: $x+5 = 0$ gives $x = -5$ .
Step 4: Exclude this value from the domain. The domain is all real numbers except $x = -5$ .

Therefore, the field of application of the equation is all real numbers except where $x = -5$ .

Thus, the domain is $x \neq -5$ .

Answer

$x\operatorname{\ne}-5$

Exercise #4

$\frac{x+y:3}{2x+6}=4$

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps to find the domain:

Step 1: Recognize that the expression $\frac{x+y:3}{2x+6}=4$ involves a fraction. The denominator $2x + 6$ must not be zero, as division by zero is undefined.
Step 2: Set the denominator equal to zero and solve for $x$ to find the values that must be excluded: $2x + 6 = 0$ .
Step 3: Solve $2x + 6 = 0$ :

$2x + 6 = 0$
$2x = -6$
$x = -3$

Step 4: Conclude that the domain of the function excludes $x = -3$ , meaning $x \neq -3$ .

Thus, the domain of the given expression is all real numbers except $x = -3$ . This translates to:

$x\operatorname{\ne}-3$

Answer

$x\operatorname{\ne}-3$

Exercise #5

$\frac{3x:4}{y+6}=6$

What is the field of application of the equation?

Video Solution

Step-by-Step Solution

To determine the field of application of the equation $\frac{3x:4}{y+6}=6$ , we must identify values of $y$ for which the equation is defined.

The denominator of the given expression is $y + 6$ . In order for the expression to be defined, the denominator cannot be zero.
This leads us to solve the equation $y + 6 = 0$ .
Solving $y + 6 = 0$ gives us $y = -6$ .
This means $y = -6$ would make the denominator zero, thus the expression would be undefined for this value.