Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=x^{2}+1, \quad g(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

The domain of \(f(x)=x^{2}+1\) is \(-\infty, \infty\). The domain of \(g(x)=\sqrt{x}\) is \(0, \infty\). The domain of the composition \(f \circ g (x)\) is \(0, \infty\).

Step 1: Determine the domain of \(f(x)\)

The domain of \(f(x)\) refers to the set of all real numbers \(x\) such that \(f(x)\) is defined. Looking at the function \(f(x)=x^{2}+1\), there are no restrictions on \(x\), which means that \(x\) can be any real number. Therefore, the domain of \(f(x)\) is \(-\infty, \infty\).

Step 2: Determine the domain of \(g(x)\)

The function \(g(x)=\sqrt{x}\), has a square root function that only allows nonnegative arguments. Therefore the domain of \(g(x)\), is only real numbers \(x\) that are greater than or equal to 0. In interval notation, this is \(0, \infty\).

Step 3: Determine the domain of \(f \circ g(x)\)

The composition of the functions \(f\) and \(g\) is \(f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2+1\). The square root of x is defined for \(x\geq 0\), and then squaring (which is part of f(x)) doesn't restrict the output from f(g(x)). So, as for \(g(x)\), the domain of \(f \circ g(x)\) is \(0, \infty\) .

Step 4: Validate with graphing utility

To confirm the results, plot the functions \(f(x)\), \(g(x)\), and \(f \circ g(x)\) on a graph. For \(f(x)\), it would be a parabola covering all x-values; for \(g(x)\) it would start from 0 and cover all x-values from there; For \(f \circ g(x)\), it should also start from 0 (like \(g(x)\)) and extend towards infinity for positive x-values.

Key Concepts

Composite Functions

Composite functions involve combining two or more functions in a manner where the output of one function becomes the input of another. In our example, we are dealing with two functions: \(f(x)=x^{2}+1\) and \(g(x)=\sqrt{x}\). The composite function is denoted as \(f \circ g(x)\), which means \(f(g(x))\). Here, we first apply \(g(x)\) and then \(f(x)\) to the result of \(g(x)\).

When identifying the domain of composite functions, we need to consider the domains of both individual functions involved. First, \(g(x)\) must be defined, and then \(f(x)\) must be able to use the output of \(g(x)\) as its input.

- **For \(g(x) \), which is \(\sqrt{x}\), the domain is \([0, \infty)\).** - **When \(g(x)\) is plugged into \(f(x)\), the result is \((\sqrt{x})^2 + 1 = x+1\), which remains defined over the interval \([0, \infty)\).**
This means the domain of the composite function \(f \circ g(x)\) is the same as that of \(g(x)\) due to this dependency.

Interval Notation

Interval notation is a concise way to represent a set of numbers, often used to describe the domain of a function. It uses brackets and parentheses:

• Round brackets \(( , )\) signify the endpoints are not included, called open intervals.
• Square brackets \([ , ]\) signify the endpoints are included, called closed intervals.

Interval notation is especially helpful in defining the domain of functions.
For example, in the expression \([0, \infty)\) for the function \(g(x) = \sqrt{x}\), the square bracket at 0 indicates that 0 is included, while the round bracket at infinity indicates that the set extends towards infinity but does not include it.

With \(f(x) = x^2 + 1\), the domain is noted as \((-\infty, \infty)\), expressing that all real numbers are included. This shows that there are no restrictions on the values \(x\) can take for \(f(x)\). Interval notation provides a quick way to visualize where these functions are defined and helps in understanding their behavior.

Graphing Utility

A graphing utility is an electronic or software tool used to visually represent mathematical functions. It simplifies understanding by displaying functions in graph form, illustrating their behavior over a range of values. In this context, we use a graphing utility to verify the domains of the functions \(f(x)\), \(g(x)\), and the composite function \(f \circ g(x)\).

Using a graphing utility:

• Plot \(f(x) = x^2 + 1\): You will see a parabolic curve extending across all x-values, verifying its domain as \((-\infty, \infty)\).
• Plot \(g(x) = \sqrt{x}\): This only starts at x=0 and extends to the right, consistent with its domain \([0, \infty)\).
• Plot \(f(g(x)) = f(\sqrt{x}) = x + 1\): This graph will also start at x=0, similar to \(g(x)\). It confirms the domain of \([0, \infty)\).

Graphing utilities make the abstract nature of functions concrete, illustrating their domains and giving immediate visual feedback.

Square Root Function

The square root function, denoted as \(g(x) = \sqrt{x}\), is a specific mathematical function that deals with finding the value which, when squared, gives the original number. Square roots tend to limit the domain of functions since they are only defined for non-negative numbers.

Key points about the square root function:

• The domain is all non-negative x-values, or \([0, \infty)\).
• The function starts at the origin \((0,0)\) and increases gradually along the positive x-axis.

This characteristic affects any composite function involving the square root function. For instance, in the composite function \(f \circ g(x)\) in our example, the domain of \(g(x)\) dictates the domain of the entire composite function because \(g(x)\) must be defined for \(f\) to receive its output. Understanding how the square root affects the domain helps to navigate more complex functions involving root expressions.