Domain of Function Practice Problems with Solutions

Understanding Inputing Values into a Function

Complete explanation with examples

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

Practice Inputing Values into a Function

Test your knowledge with 16 quizzes

Look at the following function:

$\frac{10x-3}{5x-3}$

What is the domain of the function?

Examples with solutions for Inputing Values into a Function

Step-by-step solutions included

Exercise #1

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

$\frac{9x}{4}$

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

Video Solution

Exercise #2

Look at the following function:

$\frac{5}{x}$

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

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, $x\ne0$

Video Solution

Exercise #3

Look the following function:

$\frac{1}{5x-4}$

What is the domain of the function?

Step-by-Step Solution

To determine the domain of the function $\frac{1}{5x-4}$ , we need to find the values of $x$ for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
$5x - 4 = 0$

Next, solve for $x$ :
$5x = 4$
$x = \frac{4}{5}$

The function is undefined at $x = \frac{4}{5}$ . Therefore, the domain of the function includes all real numbers except $x = \frac{4}{5}$ .

In mathematical notation, the domain is:
$x \ne \frac{4}{5}$ .

This matches choice 3 among the given options.

Answer:

$x\ne\frac{4}{5}$

Video Solution

Exercise #4

Look at the following function:

$\frac{20}{10x-5}$

What is the domain of the function?

Step-by-Step Solution

To determine the domain of the function $\frac{20}{10x-5}$ , we need to ensure that the denominator is not zero.

Step 1: Identify the denominator, which is $10x - 5$ .
Step 2: Set the denominator equal to zero and solve for $x$ . This gives us the equation:

$10x - 5 = 0$

Step 3: Add 5 to both sides of the equation:

$10x = 5$

Step 4: Divide both sides by 10 to isolate $x$ :

$x = \frac{5}{10}$

Step 5: Simplify the fraction:

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

This means that the function is undefined at $x = \frac{1}{2}$ . Therefore, the domain of the function is all real numbers except $x = \frac{1}{2}$ .

Therefore, the domain of the function is $x \ne \frac{1}{2}$ .

Answer:

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

Video Solution

Exercise #5

Given the following function:

$\frac{23}{5x-2}$

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

Step-by-Step Solution

To determine the domain of the function $\frac{23}{5x-2}$ , follow these steps:

Step 1: Identify where the function is undefined by setting the denominator equal to zero.
Equation: $5x - 2 = 0$
Step 2: Solve the equation for $x$ .

Let's perform the calculation:
Step 1: Set $5x - 2 = 0$ .

Step 2: Solve for $x$ by adding 2 to both sides:
$5x = 2$

Next, divide both sides by 5:
$x = \frac{2}{5}$

This shows that the function is undefined at $x = \frac{2}{5}$ , thus excluding this point from the domain of the function.

The domain of $\frac{23}{5x-2}$ consists of all real numbers except $x = \frac{2}{5}$ .

Therefore, the domain is expressed as $x \ne \frac{2}{5}$ .

Considering the multiple-choice options, the correct choice is:

Yes, $x\ne\frac{2}{5}$

Answer:

Yes, $x\ne\frac{2}{5}$

Video Solution

Frequently Asked Questions

Everything you need to know about Inputing Values into a Function

What is the domain of a function in simple terms?

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The domain of a function is the complete set of all possible input values (x-values) that can be substituted into the function without causing mathematical errors. Think of it as all the valid numbers you can feed into the function machine to get a meaningful output.

How do I find domain restrictions in functions?

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Look for these common restrictions: 1) Division by zero - set denominators ≠ 0, 2) Square roots of negative numbers - set expressions under square roots ≥ 0, 3) Logarithms of non-positive numbers - set arguments > 0. Solve these inequalities to find the domain.

What happens when I substitute values into a function?

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When you substitute a value for the variable in a function, you replace every instance of that variable with the given number. Then you perform the mathematical operations in the correct order to calculate the function's output value.

Why can't some values be in a function's domain?

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Certain values are excluded from domains because they create undefined mathematical situations. The most common reasons are division by zero, taking square roots of negative numbers, or taking logarithms of zero or negative numbers.

How do I write domain in interval notation?

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Use parentheses ( ) for values not included and brackets [ ] for included values. For example: (-∞, 3) ∪ (3, ∞) means all real numbers except 3. Use ∪ to combine separate intervals when there are gaps in the domain.

What's the difference between domain and range of a function?

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Domain refers to all possible input values (x-values) you can put into a function, while range refers to all possible output values (y-values) the function can produce. Domain is what goes in, range is what comes out.

Can a function have an empty domain?

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Technically yes, but such functions are not useful in practice. For example, f(x) = √(x² + 1) has domain of all real numbers, but f(x) = √(x² + 1) where x² + 1 < 0 would have an empty domain since x² + 1 is always positive.

How do I check if a value is in a function's domain?

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Substitute the value into the function and check if it creates any mathematical errors. If you can calculate a real number output without dividing by zero or taking square roots of negatives, then the value is in the domain.

Continue Your Math Journey

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Domain of Function Practice Problems with Solutions

Understanding Inputing Values into a Function

Inputing Values into a Function

Understanding Variables and Substitution

For example

Practice Inputing Values into a Function

Examples with solutions for Inputing Values into a Function

Frequently Asked Questions

What is the domain of a function in simple terms?

How do I find domain restrictions in functions?

What happens when I substitute values into a function?

Why can't some values be in a function's domain?

How do I write domain in interval notation?

What's the difference between domain and range of a function?

Can a function have an empty domain?

How do I check if a value is in a function's domain?

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