Problem 72 - Precalculus Essentials

Question

Find; a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\sqrt{x}, g(x)=x-3$$

Step-by-Step Solution

Answer

a. The composite of the function is \( f(g(x))=\sqrt{x-3} \) \n b. The domain of the function \( f(g(x)) \) is \( [3,+\infty) \)

Step 1: Find the Composition of Function

To find the composite function \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \). That would result in \( f(g(x)) \). Given \( f(x)=\sqrt{x} \), and \( g(x)=x-3 \), then \( f(g(x)) = \sqrt{g(x)} = \sqrt{x-3} \)

Step 2: Find the Domain of the Composite Function

The domain of \( f(g(x)) = \sqrt{x-3} \) is the set of all real numbers \( x \) for which \( x-3 \geq 0 \). Therefore, x must be greater than or equal to 3. Hence, the domain of \( f(g(x)) \) is \( [3, +\infty) \)

Key Concepts

Domain of a Function

The domain of a function is a crucial concept in mathematics that refers to all possible input values (or "independent variables") that a function can accept without causing any mathematical inconsistencies. Understanding the domain helps ensure that the function behaves as expected for these values.

In more formal terms, it's the set of all values of \(x\) for which the function \(f(x)\) is defined. For example, if you have a function such as \(f(x) = \frac{1}{x}\), the domain excludes \(x = 0\) because division by zero is undefined.

Here's a quick guide to determine the domain:

• For polynomial functions, the domain is all real numbers.
• For a rational function \(\frac{p(x)}{q(x)}\), exclude values that make \(q(x) = 0\).
• For functions involving square roots, exclude values that result in a negative number inside the square root (since the square root of a negative number is not defined in the real number system).

Recognizing these patterns will help you determine the domain of a wider variety of functions.

Square Root Function

The square root function takes the form \(f(x) = \sqrt{x}\), and it is one of the simplest types of functions involving radicals. Its behavior is straightforward: it produces an output that, when multiplied by itself, results in the input value. The graph of this function is a curve that starts at the origin (0,0) and gradually increases, only existing in the first quadrant.

As such, the domain of the square root function \(f(x) = \sqrt{x}\) is all non-negative numbers. This is because the square root of a negative number does not produce a real number result.

• For example, \(f(x) = \sqrt{x}\) is defined for \(x \geq 0\).
• If you encounter \(f(x) = \sqrt{x-3}\), then \(x-3\) must be \(\geq 0\), translating to \(x \geq 3\).

These conditions ensure that the square root function is operating within the real numbers, keeping the calculations valid and meaningful.

Composite Function Domain

When dealing with composite functions, such as \((f \/\circ \/g)(x)\), understanding their domain requires looking at the individual functions involved. Essentially, a composite function is formed by applying one function to the results of another function.

To determine the domain of \((f \/\circ \/g)(x)\), you need to consider:

• The domain of the "inside" function - \(g(x)\). This is where \(g(x)\) is first computed.
• The resulting values of \(g(x)\) must lie within the domain of the "outside" function \(f(x)\).

For example, if \(f(x) = \sqrt{x}\) and \(g(x) = x-3\), \((f \/\circ \/g)(x) = \sqrt{x-3}\). In this case, \(x-3\) must be non-negative, leading to \(x \geq 3\). Thus, the domain of \((f \/\circ \/g)(x)\) is \([3, \infty)\). Ensuring that each step aligns with these principles guarantees that the composite function \((f \/\circ \/g)(x)\) is defined and meaningful over its domain.