Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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1:16 minutes

Problem 41

Textbook Question

In Exercises 31–50, find ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Verified step by step guidance

Step 1: Understand the problem. We need to find the function \( (f-g)(x) \), which is the difference between \( f(x) \) and \( g(x) \).

Step 2: Write the expression for \( (f-g)(x) \). This is given by \( (f-g)(x) = f(x) - g(x) \).

Step 3: Substitute the given functions into the expression. \( f(x) = 2 + \frac{1}{x} \) and \( g(x) = \frac{1}{x} \), so \( (f-g)(x) = \left(2 + \frac{1}{x}\right) - \frac{1}{x} \).

Step 4: Simplify the expression. Combine like terms: \( (f-g)(x) = 2 + \frac{1}{x} - \frac{1}{x} = 2 \).

Step 5: Determine the domain. The domain of \( f(x) \) and \( g(x) \) is all real numbers except \( x = 0 \) because division by zero is undefined. Therefore, the domain of \( (f-g)(x) \) is also all real numbers except \( x = 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Subtraction

Function subtraction involves taking two functions, f(x) and g(x), and creating a new function, ƒ-g, defined as ƒ(x) - g(x). This operation requires combining the outputs of both functions for the same input value, which can lead to new expressions that may have different properties than the original functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like f(x) = 2 + 1/x and g(x) = 1/x, the domain excludes values that make the denominator zero, as these would result in undefined outputs.

Combining Domains

When subtracting two functions, the domain of the resulting function ƒ-g is determined by the intersection of the domains of f(x) and g(x). This means that any x-value that is not in the domain of either function cannot be included in the domain of the new function, ensuring that ƒ-g is defined for those inputs.