Problem 46 - 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 is \([0, 5]\).

Step 1: Define Domain Range

The domain of a function refers to all the possible input values (or x-values) that the function can accept. In this case, the function is given as \( f(x) = x^2 + 1 \) with the restriction \( 0 \leq x \leq 5 \). This means the domain is predefined by this condition.

Step 2: Analyze Implicit Domain Restrictions

Considering the function \( f(x) = x^2 + 1 \), there are no further restrictions on the domain imposed by the function itself, such as division by zero or square roots of negative numbers. This confirms that all values within the given range \( 0 \leq x \leq 5 \) are valid inputs.

Step 3: State the Domain

The only restriction on the domain arises from the given interval \( 0 \leq x \leq 5 \). Therefore, the domain of the function \( f(x) = x^2 + 1 \) is \([0, 5]\).

Key Concepts

Domain Restrictions

When we talk about the domain of a function, we refer to all the possible input values (or x-values) that the function can accept without encountering any mathematical problems. In simpler terms, it's about figuring out the "allowed" x-values for a given function. Sometimes, functions come with built-in restrictions such as:

• Division by zero, which is undefined and must be avoided,
• Square roots of negative numbers in the set of real numbers.

In the case of the function \( f(x) = x^2 + 1 \), however, there aren't any inherent problematic conditions like division by zero. This specific problem provides a predefined range \( 0 \leq x \leq 5 \), meaning the acceptable x-values are those within this interval.Thus, the domain restrictions originate solely from this predefined condition itself, simplifying our task to only consider values between 0 and 5, inclusive.

Function Analysis

Function analysis involves breaking down the given function to understand its behavior, properties, and limitations. In our function, \( f(x) = x^2 + 1 \), we need to understand what happens to the output values as the input varies.Given the expression \( x^2 + 1 \), this function is a simple quadratic function. Quadratic functions typically result in a parabolic shape. The expression \( x^2 + 1 \) means you start with the square of an input (x), then you add one. It increases throughout its domain, indicating no dips or undefined sections. The beautiful part about this function is its simplicity: it naturally handles any real x-value without worries like undefined sections.For our task, the focus is not on managing any complex behavior that the function could have. Instead, we primarily consider only the provided range \( 0 \leq x \leq 5 \) because there are no other implicit constraints affecting input values.

Input Values

The term "input values" refers to the numbers you can use for x in the function. They represent the potential values that can be substituted into the function’s formula. These are governed by the concept of the domain.For \( f(x) = x^2 + 1 \) with the specified range \( 0 \leq x \leq 5 \):

• The input values start at 0 and go up to 5,
• This includes all numbers in between such as 1, 2, 3.5, etc., and even non-integers like 2.7,
• Every valid input conforms to the simple constraint provided from the start.

It's essential to note that the input values are the driving force of a function, as they are what you manipulate to observe how the function behaves. As such, recognizing and correctly identifying these values is crucial in understanding and applying functions in practical and theoretical mathematics.