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

Start practice

Test yourself on domain of a function!

Look at the following function:

$\frac{5}{x}$

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

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

On the Tutorela website, you will find a variety of articles with interesting explanations about mathematics

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

Exercise #1

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 #2

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 #3

Look at the following function:

$\frac{10x}{\frac{1}{2}}$

What is the domain of the function?

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

Step 1: Simplify the expression.

Given the function:

$f(x) = \frac{10x}{\frac{1}{2}}$

We can simplify this expression by multiplying by the reciprocal of the denominator:

$f(x) = 10x \times 2 = 20x$

Step 2: Determine the domain.

Since $f(x) = 20x$ is a linear function, it is defined for all real numbers. There are no restrictions on $x$ since no division by zero or any undefined operations are present.

Conclusion: The domain of the function is all real numbers. This corresponds to choice :

All real numbers

Therefore, the domain of the function is all real numbers.

Answer

All real numbers

Exercise #4

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

What is the domain of the equation above?

Video Solution

Step-by-Step Solution

To find the domain of the given function $22\left(\frac{2}{x} - 1\right) = 30$ , follow these steps:

Identify critical terms: The term $\frac{2}{x}$ is undefined when $x = 0$ because division by zero is undefined.
We need to exclude $x = 0$ from the domain to ensure the function remains defined.
The correct domain for the equation is all real numbers except $x = 0$ .

Thus, the domain of the equation is $x \neq 0$ .

Therefore, the solution to the problem is $x \neq 0$ .

Answer

x≠0

Exercise #5

$2x+\frac{6}{x}=18$

What is the domain of the above equation?

Video Solution

Step-by-Step Solution

To solve this problem and find the domain for the expression $2x + \frac{6}{x}$ , we apply the following steps:

Step 1: Identify when the fraction $\frac{6}{x}$ is undefined. This occurs when the denominator $x$ equals zero.
Step 2: To find the restriction, set the denominator equal to zero: $x = 0$ .
Step 3: Solve for $x$ to find the values excluded from the domain. Here, $x \neq 0$ .

Since $\frac{6}{x}$ is undefined for $x = 0$ , the value $x = 0$ must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.