Problem 40 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=x^{2}+1, \quad 0 \leq x \leq 5 $$

Step-by-Step Solution

Answer

The domain of the function is [0, 5].

Step 1: Understand the Function's Equation

The function given is \( f(x) = x^2 + 1 \). This is a basic polynomial function.

Step 2: Analyze Polynomial Nature

Polynomial functions like \( x^2 + 1 \) are defined for all real numbers because evaluating a real number inside a polynomial does not involve division by zero or roots of negative numbers.

Step 3: Consider Given Constraints

The problem specifies constraints, \( 0 \leq x \leq 5 \). This means the function is only defined for this range of \( x \) values, restricting the otherwise unlimited domain.

Step 4: Determine the Domain

Taking into account both the polynomial nature of the function and the specified constraints, the domain consists of all \( x \) values within the interval [0, 5].

Key Concepts

Understanding Polynomial Functions

A polynomial function is a type of mathematical expression that involves terms composed of variables raised to whole number exponents, and coefficients as constants. In simple terms, if you take a variable, multiply it by itself several times, and then add or subtract other similar combinations, you get a polynomial function.
For example, the function given in the exercise, \( f(x) = x^2 + 1 \), is a polynomial function. Here, the term \( x^2 \) involves the variable \( x \) raised to the power of 2, and \(+1\) acts as a constant.
What's special about polynomial functions is their flexibility and smoothness. Typically, they are continuous and well-behaved, which means:

• They don't have sharp corners.
• They are defined for all real numbers, unless specific restrictions are made.
• They are easy to differentiate and integrate, making them useful in calculus.

Because of these properties, polynomial functions are widely used to model real-world scenarios where smooth curves represent behaviors or trends.

Exploring Domain Restrictions

The domain of a function comprises all the possible values that you can plug into a function to get valid outputs. While polynomial functions like \( x^2 + 1 \) are naturally defined for all real numbers, domain restrictions can be introduced to limit this range.
In our exercise, even though the function has the potential to accept any real number input, the specified constraint \( 0 \leq x \leq 5 \) restricts the domain. This means the function is only defined for \( x \) values that fall within this range.

• Purpose of Restrictions: Domain restrictions are often introduced to model real-world conditions or scenarios where a function only makes sense within certain intervals.
• Example: The height of an object thrown in the air can be modeled by a polynomial, but the object only exists during the flight time, creating a domain restriction.

Understanding domain restrictions helps in accurately applying mathematical functions to practical situations by focusing only on the relevant range of values.

Role of Inequalities in Domain

Inequalities in domain help specify which values are permissible for input in a function, especially when the natural domain might otherwise be broader. These inequalities provide boundaries, much like a fence, to enclose the domain.
In the function \( f(x) = x^2 + 1 \) with the constraint \( 0 \leq x \leq 5 \), the inequalities \( 0 \leq x \) and \( x \leq 5 \) jointly define the domain. They tell us that the function takes values in the range from 0 to 5, inclusive.

• Importance of Inequalities: They ensure that only valid computations occur within specified limits, preventing undefined behaviors or nonsensical results.
• How to Use Them: When analyzing inequalities, it's important to note whether endpoints are included. The symbols \( \leq \) or \( \geq \) signify inclusion of the boundary values.

This particular use of inequalities plays a crucial role in ensuring the function is applied correctly and appropriately, in accordance with the conditions of the problem or scenario being modeled.