Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Determine the domain of (a) \(f\), (b) \(g\), and (c) \(f \circ g\). \(f(x)=\frac{1}{x^{2}}, \quad g(x)=x-2\)

Step-by-Step Solution

Answer

The domain of \(f(x) = \frac{1}{x^{2}}\) is \(x \in \mathbb{R}\), \(x ≠ 0\). The domain of \(g(x)= x-2\) is \(x \in \mathbb{R}\). The domain of the composite function \(f(g(x))\) is \(x \in \mathbb{R}\), \(x ≠ 2\).

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

The function \(f(x) = \frac{1}{x^{2}}\) is defined for all real numbers, except \(x = 0\). So, the domain of \(f(x)\) is \(x \in \mathbb{R}\), \(x ≠ 0\).

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

The function \(g(x)= x-2\) is a linear function and is defined for all real numbers. So, the domain of \(g(x)\) is \(x \in \mathbb{R}\).

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

To find the domain of the composite function \(f(g(x))\), we must consider the domain of \(f\) and the output of \(g\). As discussed in Step 1, \(f\) is undefined when \(x=0\), and in Step 2 we found that \(g(x)\) can take all real values. We need to find \(x\) for which \(g(x) = 0\), because for those \(x\), \(f(g(x))\) will be undefined. Solving \(x-2 = 0\) we get \(x = 2\). So, the domain of the composite function \(f(g(x))\) is \(x \in \mathbb{R}\), \(x ≠ 2\).

Key Concepts

Composite Functions

Composite functions combine two functions to create a new one. Think of it like nesting one function inside another. This is often written as \( f(g(x)) \), read as "f of g of x". It's like applying function \( g \) first and then taking the result to function \( f \).
The domain of a composite function depends on the domains of both functions involved. Specifically, for \( f(g(x)) \), the output values of \( g(x) \) must be suitable inputs for \( f(x) \). It's important to ensure that the intermediate step—the output of \( g(x) \)—falls within the domain of \( f \).
To find the domain of a composite function, follow these steps:

• Identify the domain of the inside function \( g(x) \).
• Ensure the outputs of \( g(x) \) fit within the domain of the outside function \( f(x) \).

In our exercise, we need to make sure that \( g(x) \) doesn't output a value that makes \( f \) undefined.

Real Numbers

Real numbers form the backbone of many mathematical concepts. They include all numbers that can be found on the number line, encompassing both rational and irrational numbers. So, every number we can think of, from integers like \( -5 \) and 2, to fractions like \( \frac{1}{3} \), and irrational numbers like \( \sqrt{2} \), are real numbers.
When referring to the domain of functions, real numbers (\( \mathbb{R} \)) provide a broad set of values that functions can "move around in". For a function like \( g(x) = x - 2 \), which is defined for all real numbers, it means you can practically plug in any real number value to see how \( g \) transforms it.
In exercises involving functions, it's often about understanding where these real numbers can go—not just in one function, but when they're threaded through a series of functions, like in composite functions.

Linear Function

Linear functions are among the simplest types of functions. These functions have no powers, roots, or complicated patterns—just a plain straight slope represented as \( f(x) = mx + b \). Think of the graph as a straight line where \( m \) is the slope, indicating the line's steepness, and \( b \) is the y-intercept, telling us where the line crosses the y-axis.
These functions are straightforward, and importantly, unlike other functions like \( \frac{1}{x^2} \), they are defined everywhere on the number line. As a result, their domain is all real numbers (\( x \in \mathbb{R} \)).
In our exercise, \( g(x) = x - 2 \) is a linear function. This means it smoothly marches through every real number without getting tripped up by exceptions.