Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=x^{2}+3, \quad g(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

The domain of \(f\) is \(x \in \mathbb{R}\), the domain of \(g\) is \(x \geq 0\), and the domain of \(f \circ g\) is \(x \geq 0\).

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

The function \(f(x)=x^{2}+3\) is a quadratic function, it is defined for all real numbers. Thus, the domain of \(f\) is \(x \in \mathbb{R}\).

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

The function \(g(x)=\sqrt{x}\) is a square root function. The square root function is only defined for numbers equal to or greater than zero. Therefore, the domain of \(g\) is \(x \geq 0\).

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

The composite function is \(f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2 + 3 = x + 3\). Since we are taking the square of \(g(x)\) , the domain should at least be all non-negative real numbers, which is the original domain of \(g\). But given the definition of \(f\), adding 3 to all x also applies to all real numbers. As such, the domain for \(f(g(x))\) or \(f \circ g\) is \(x \geq 0\), same as the domain of \(g\).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (x-values) that allow the function to work without any mathematical errors. Every function has its own domain, determined by the type of operation involved. In the exercise above, we deal with two functions: a quadratic function and a square root function, each having unique domains because of their fundamental properties.
For a quadratic function, like \( f(x) = x^2 + 3 \), we have a polynomial. Polynomial functions are simple in terms of domains. They can accept any real number as input. Thus, the domain of the quadratic function \( f \) is every real number, or \( x \in \mathbb{R} \).
On the other hand, the square root function \( g(x) = \sqrt{x} \) is a bit more restrictive. The square root function requires the radicand (the number under the square root) to be non-negative to avoid imaginary numbers. Therefore, the domain of \( g \) is all real numbers where \( x \geq 0 \). This difference in domains illustrates how distinct types of functions impose various rules on possible inputs.

Quadratic Functions

Quadratic functions are a type of polynomial that have the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). They produce a parabolic curve when graphed, either opening upwards or downwards based on the sign of \( a \).
Key characteristics of quadratic functions include:

• The vertex, which is the highest or lowest point on the graph.
• The axis of symmetry, a vertical line passing through the vertex, given by the formula \( x = -\frac{b}{2a} \).
• The roots or zeros, the values of \( x \) that make the function's value zero.

Although the function \( f(x) = x^2 + 3 \) doesn't have real roots (since \( x^2 + 3 = 0 \) results in no real solutions), its domain remains all real numbers. Unlike square root functions, quadratic functions don't inherently limit the domain since they don't involve operations that require restrictions to avoid imaginary numbers.

Square Root Functions

Square root functions, such as \( g(x) = \sqrt{x} \), involve applying the square root operation, which introduces specific domain constraints. These functions are defined by the need to keep the expression under the square root non-negative.
The primary features of square root functions are:

• A domain restricted to non-negative numbers (\( x \geq 0 \)), because a negative value under the square root would result in imaginary numbers.
• A range that begins at zero and extends to positive infinity, reflecting the outputs produced only by non-negative inputs.

The graph of a square root function is a curve starting at the origin (or another starting point based on added constants) and increases slowly to the right. This form highlights the unique behavior of square root functions compared to other types, emphasizing why understanding domain restrictions is crucial when working with these functions.