Problem 9 - Precalculus

Question

Find an expression for \(f(x)\) and state its domain. \(f\) is a function that takes a real number \(x\) and performs the following three steps in the order given: (1) take the square root; (2) make the quantity the denominator of a fraction with numerator \(4 ;\) (3) subtract 13 .

Step-by-Step Solution

Answer

The expression for \(f(x)\) is \(f(x) = \frac{4}{\sqrt{x}} - 13\), and its domain is \((0, \infty)\).

Step 1: Identify the Mathematical Operations

The function \(f(x)\) is defined by applying a series of operations to \(x\). Begin by taking the square root of \(x\). This operation can be expressed as \(\sqrt{x}\).

Step 2: Create the Fraction

Next, according to the instructions, make \(\sqrt{x}\) the denominator of a fraction with numerator 4. The expression becomes \(\frac{4}{\sqrt{x}}\).

Step 3: Subtract a Constant

Subtract 13 from the fraction \(\frac{4}{\sqrt{x}}\). Therefore, the expression for \(f(x)\) is \(f(x) = \frac{4}{\sqrt{x}} - 13\).

Step 4: Determine the Domain of the Function

The square root function \(\sqrt{x}\) requires that \(x\geq 0\) to have real values. The denominator \(\sqrt{x}\) cannot be zero as division by zero is undefined, which imposes \(x > 0\). Therefore, the domain of \(f(x)\) is all positive real numbers: \((0, \infty)\).

Key Concepts

Domain of a Function

When dealing with functions, one key aspect is identifying their domain. The domain of a function refers to the set of all possible input values (typically represented by \(x\)) for which the function is defined. In simple terms, the domain tells us what values we can plug into the function without encountering any issues such as division by zero or taking the square root of a negative number.

For our function \(f(x) = \frac{4}{\sqrt{x}} - 13\), we need to analyze the two operations involved: the square root and the fraction.- The square root \(\sqrt{x}\) is only defined for non-negative numbers, so \(x\geq 0\) must be true.- However, because \(\sqrt{x}\) is in the denominator, it cannot be zero (as division by zero is undefined). Therefore, we need \(x > 0\).

Combining these two constraints, the domain of our function \(f(x)\) is all positive real numbers, represented as \((0, \infty)\). This ensures that each operation in the function is performed legally and produces a real number.

Square Root Function

Square root functions are an interesting and important type of mathematical operation. They involve taking the square root of a number or expression, symbolized by \(\sqrt{\cdot}\). The square root function transforms a given input into another value whose square is equal to the input.

The key characteristics of a square root function include:

• Defined only for non-negative numbers: This means \(x\geq 0\).
• The result is always non-negative: For any \(x\) within the domain, \(\sqrt{x} \geq 0\).

In the function \(f(x) = \frac{4}{\sqrt{x}} - 13\), the operation \(\sqrt{x}\) serves as a crucial step, determining the denominator of a fraction. Since it only accepts non-negative inputs, it guides the expression's domain to be all positive real numbers.

Fraction Operations

Fraction operations entail dealing with numerators and denominators, and they follow specific rules that must be considered to solve or simplify expressions involving fractions. In basic fractional operations:

• The numerator is the top part of the fraction, determining how many parts of the whole are being considered.
• The denominator, the bottom part, indicates the total number of equal parts the whole is divided into.

For the given function \(f(x) = \frac{4}{\sqrt{x}} - 13\), the square root \(\sqrt{x}\) becomes the denominator, which needs careful handling.

Key aspects to remember include:- A denominator must never be zero, as division by zero is undefined.- To simplify or correctly use a fraction, ensure that the denominator remains within the set constraints, like staying positive in the case of our function.

By making these considerations, one can confidently approach, simplify or manipulate fractions in various mathematical contexts.